# Formal first integrals and higher variational equations

Sergi Simon

School of Mathematics and Physics, University of Portsmouth
Lion Gate Bldg, Lion Terrace Portsmouth PO1 3HF, UK
<EMAIL_ADDRESS>
###### Abstract

The question of how Algebra can be used to solve dynamical systems and
characterize chaos was first posed in a fertile mathematical context by
Ziglin, Morales, Ramis and Simó using differential Galois theory. Their study
was aimed at first-order, later higher-order, variational equations of
Hamiltonian systems. Recent work by this author formalized a compact yet
comprehensive expression of higher-order variationals as one infinite linear
system, thereby simplifying the approach. More importantly, the dual of this
linear system contains all information relevant to first integrals, regardless
of whether the original system is Hamiltonian. This applicability to formal
calculation of conserved quantities is the centerpiece of this paper,
following an introduction to the requisite context. Three important examples,
namely particular cases of Dixon’s system, the SIR epidemiological model with
vital dynamics and the Van der Pol oscillator, are tackled, and explicit
convergent first integrals are provided for the first two.

Keywords. Integrability, Ziglin-Morales-Ramis-Simó theory, formal calculus,
chaos, Dixon’s system, SIR epidemiological model.

2000 Mathematics Subject Classification: 34A05, 37C79, 37J99, 37J30, 34M15,
34C28, 37C10, 15A69, 16W60, 13F25, 37N25.

##  1 Introduction

Given an arbitrary dynamical system, the formulation of its higher variational
equations as a linear (infinite) system $\left({\mathrm{LVE}^{\star}}\right)$
has shown potential to make strong inroads in the study of integrability [34,
35]. The study, part of which we will explain more in detail in §1.1, is
twofold:

  1. (i)

on one hand, the original set of equations $\left({\mathrm{LVE}}\right)$ is
amenable to the Ziglin-Morales-Ramis-Simó non-integrability framework whenever
the system is _Hamiltonian_ and the first integrals whose existence is
obstructed are _meromorphic_ ([27, 28, 29, 30] and a long assorted array of
references derived therefrom, including [34, 35]); there is further, as of yet
unfinished study in the direction of situations where the system is not
Hamiltonian ([16, 19, 20]).

  2. (ii)

on the other, the _dual_ system $\left({\mathrm{LVE}}\right)^{\star}$ has jets
of formal first integrals among its solutions ([3, 34]), which entails the
possibility to furnish first integrals instead of finding obstructions to
them; the advantage of this second approach is that the original system need
not be Hamiltonian, and the formal first integrals, if convergent need not be
meromorphic. The only difficulties in this case are computational in nature,
namely in the context of resummation techniques.

In the present work we will exploit the second item (ii), namely by describing
a method to automatically produce Taylor terms of formal first integrals by
way of an automatic, easily recursified sequence of quadratures. First we
recount the minimal background exposition necessary, then we present the main
results in §2 and finally we apply it to two simple examples to test its
accuracy and usefulness as well as make novel statements about the
integrability of the examples themselves.

###  1.1 The algebraic study of integrability

####  1.1.1 Basics

Let $\psi\left({t,\cdot}\right)$ be the flow and
${\phi}\left(t\right)=\psi\left({t,\mbox{\boldmath${x}$\unboldmath}}\right)$ a
particular solution of a given autonomous system

$\dot{{\mbox{\boldmath${z}$\unboldmath}}}=X\left(\mbox{\boldmath${z}$\unboldmath}\right),\qquad
X:{\mathbb{C}}^{m}\to{\mathbb{C}}^{m},$ (1)

respectively. The variational system of (1) along ${\phi}$ has
$\frac{\partial}{\partial{z}}\psi\left({t,{\phi}}\right)$ as a fundamental
matrix:

$\dot{Y}=A_{1}Y,\quad A_{1}\left({t}\right):=\left.\frac{\partial
X}{\partial\mbox{\boldmath${z}$\unboldmath}}\right|_{{\small{z}={\phi}\left({t}\right)}}\in\mathrm{Mat}_{n}\left({K}\right),$
($\mathrm{VE}_{\phi}$)

$K={\mathbb{C}}\left({\mbox{\boldmath${\phi}$\unboldmath}}\right)$ being the
smallest differential field ([36, Def. 1.1]) containing all entries of
$\mbox{\boldmath${\phi}$\unboldmath}\left({t}\right)$.
$\frac{\partial^{k}}{\partial{\small\mbox{\boldmath${z}$\unboldmath}}^{k}}\psi\left({t,{\phi}}\right)$
are multilinear $k$-forms appearing in the Taylor expansion of the flow along
${\phi}$:

$\psi\left({t,{z}}\right)=\psi\left({t,{\phi}}\right)+\sum_{k=1}^{\infty}\frac{1}{k!}\frac{\partial^{k}\psi\left({t,{\phi}}\right)}{\partial{z}^{k}}\left\\{{{z}-{\phi}}\right\\}^{k};$
(2)

$\partial^{k}_{z}\psi\left({t,{\phi}}\right)$ also satisfy an echeloned set of
systems, depending on the previous $k-1$ partial derivatives and usually
called order-$k$ variational equations $\mathrm{VE}_{\phi}^{k}$ ([30, p.
859-861], [34, Corollary 3]). Thus we have, (1) given, a _linear_ system
${{\mathrm{VE}_{\phi}}}=:{{\mathrm{VE}_{\phi}^{1}}}=:{{\mathrm{LVE}_{\phi}^{1}}}$
and a family of _non-linear_ systems
$\left\\{{{{\mathrm{VE}_{\phi}^{k}}}}\right\\}_{k\geq 2}$.

[34] presented an explicit _linearized_ version ${\mathrm{LVE}_{\phi}^{k}}$,
$k\geq 1$, by means of symmetric products $\odot$ of finite and infinite
matrices based on already-existing definitions by Bekbaev, e.g. [4]. This was
done in preparation for the Ziglin-Morales-Ramis-Simó (ZMRS) theoretical
framework based on monodromy and differential Galois groups [27, 36, 38], but
has other consequences as well. More specifically, our outcomes in [34] have
two applications for system (1), _Hamiltonian or not_ :

  * •

full structure of ${\mathrm{VE}_{\phi}^{k}}$ and ${\mathrm{LVE}_{\phi}^{k}}$,
i.e. _recovering the flow_ , which underlies the ZMRS theoretical corpus in
practicality, albeit as a tool rather than as a goal;

  * •

a byproduct is the full structure of dual systems
$\left({{\mathrm{LVE}_{\phi}^{k}}}\right)^{\star}$, i.e. _recovering formal
first integrals of ( 1)_ in ways which simplified earlier results in [3]
significantly.

As said in the introduction, results in the present paper are based on the
second of these applications. For examples of the first application, see [34,
§6] or the bulk of [35]. See also [22, 23] for examples where the non-
linearized $\mathrm{VE}_{k}$ were used.

###  1.2 Symmetric products, powers and exponentials

######  Notation 1.2.1.

The conventions listed below were already introduced in [3, 34, 35]:

  1. 1.

_Multi-index modulus, arithmetic, order and lexicographic order_ : for
$\mathbf{i}=\left(i_{1},\ldots,i_{n}\right)\in{\mathbb{Z}}_{\geq 0}^{n}$,
$i=\left|\mathbf{i}\right|:=\sum_{k}i_{k}$; addition and subtraction are
defined entrywise as usual; $\mathbf{i}\leq\mathbf{j}$ means $i_{k}\leq j_{k}$
for every $k\geq 1$; $\mathbf{i}<_{\mathrm{lex}}\mathbf{j}$ if
$i_{1}=j_{1},\ldots,i_{k-1}=j_{k-1}$ and $i_{k}<j_{k}$ for some $k\geq 1$.

  2. 2.

Whenever such derivation is possible, we define the lexicographically sifted
differential of $F\left({z_{1},\dots,z_{m}}\right)$ of order $m$ as the row
vector

$F^{(m)}\left(\mbox{\boldmath${z}$\unboldmath}\right):=\mathrm{lex}\left(\frac{\partial^{m}F}{\partial
z^{i_{1}}_{1}\ldots\partial
z^{i_{n}}_{n}}\left(\mbox{\boldmath${z}$\unboldmath}\right)\right),$ (3)

where $\left|{\mathbf{{i}}}\right|=m$ and entries are ordered as per
$<_{\mathrm{lex}}$ on multi-indices. For instance, for $n=2$ the first two
differentials would be

$F^{(1)}=\left({\begin{array}[]{ccc}\frac{\partial F}{\partial
z_{1}}&\frac{\partial F}{\partial z_{2}}&\frac{\partial F}{\partial
z_{3}}\end{array}}\right),\quad
F^{(2)}=\left({\begin{array}[]{cccccc}\frac{\partial^{2}F}{\partial
z_{1}^{2}}&\frac{\partial^{2}F}{\partial
z_{1}z_{2}}&\frac{\partial^{2}F}{\partial
z_{1}z_{3}}&\frac{\partial^{2}F}{\partial
z_{2}^{2}}&\frac{\partial^{2}F}{\partial
z_{2}z_{3}}&\frac{\partial^{2}F}{\partial z_{3}^{2}}\end{array}}\right).$

  3. 3.

We define $d_{n,k}:=\binom{n+k-1}{n-1},\;D_{n,k}:=\sum_{i=1}^{k}d_{n,i}.$ It
is easy to check there are $d_{n,k}$ $k$-ples of integers in
$\left\\{{1,\dots,n}\right\\}$, and just as many homogeneous monomials of
degree $n$ in $k$ variables.

  4. 4.

Given integers $k_{1},\dots,k_{n}\geq 0$, we define the usual multinomial
coefficient as

$\binom{k_{1}+\dots+k_{n}}{k_{1},\dots,k_{n}}:=\binom{k_{1}+\dots+k_{n}}{\mathbf{{k}}}:=\frac{\left({k_{1}+\dots+k_{n}}\right)!}{k_{1}!k_{2}!\cdots
k_{n}!}.$

For a multi-index $\mathbf{{k}}\in{\mathbb{Z}}_{\geq 0}^{n}$, define
$\mathbf{{k!}}:=k_{1}!\cdots k_{n}!$. For any two such
$\mathbf{{k}},\mathbf{{j}}$, we define

$\binom{\mathbf{k}}{\mathbf{p}}:=\frac{k_{1}!k_{2}!\cdots
k_{n}!}{p_{1}!p_{2}!\cdots
p_{n}!\left({k_{1}-p_{1}}\right)!\left({k_{2}-p_{2}}\right)!\cdots\left({k_{n}-p_{n}}\right)!}=\binom{k_{1}}{p_{1}}\binom{k_{2}}{p_{2}}\cdots\binom{k_{n}}{p_{n}},$
(4)

and the multi-index counterpart to the multinomial,
$\binom{\mathbf{{k}}_{1}+\dots+\mathbf{{k}}_{m}}{\mathbf{{k}}_{1},\dots,\mathbf{{k}}_{m}}:=\frac{\left({\mathbf{{k}}_{1}+\dots+\mathbf{{k}}_{n}}\right)!}{\mathbf{{k}}_{1}!\mathbf{{k}}_{2}!\cdots\mathbf{{k}}_{n}!}.$

The compact formulation called for by Notation 1.2.1 (3) was achieved in [34]
through an operation $\odot$ that had already been defined by Bekbaev (e.g.
[4, 5]) and was systematized using basic categorical properties of the tensor
product. Let $K$ be a field and $V$ a $K$-vector space. Let
${\mathrm{Sym}}^{r}V$ be the symmetric power of $V$. We write
$\mbox{\boldmath${w}$\unboldmath}_{1}\odot\mbox{\boldmath${w}$\unboldmath}_{2}$
for equivalence classes of tensor products of these vectors. Hence, product
$\odot$ operates exactly like products of homogeneous polynomials in several
variables.

######  Notation 1.2.2.

When dealing with matrix sets, we will use super-indices and subindices:

  1. 1.

The space of $\left({i,j}\right)$-matrices
${\mathrm{Mat}}_{m,n}^{i,j}\left({K}\right)$ is defined equivalently as the
set of $d_{m,i}\times d_{n,j}$ matrices with entries in $K$, or vector space
${\mathrm{Hom}}_{K}\left({{\mathrm{Sym}}^{j}K^{m};{\mathrm{Sym}}^{i}K^{n}}\right)$.

  2. 2.

It is clear from the above that ${\mathrm{Mat}}_{n}^{0,0}\left({K}\right)$ is
the set of all scalars $\alpha\in K$ and
${\mathrm{Mat}}_{n}^{0,k}\left({K}\right)$ (resp.
${\mathrm{Mat}}_{n}^{k,0}\left({K}\right)$) is made up of all row (resp.
column) vectors whose entries are indexed by $d_{n,k}$ lexicographically
ordered $k$-tuples.

  3. 3.

Reference to $K$ may be dropped and notation may be abridged if dimensions are
repeated or trivial, e.g.
${\mathrm{Mat}}_{n}^{i,j}:={\mathrm{Mat}}_{n,n}^{i,j}$,
${\mathrm{Mat}}_{m,n}^{i}:={\mathrm{Mat}}_{m,n}^{i,i}$,
${\mathrm{Mat}}_{n}:={\mathrm{Mat}}_{n}^{1}$, etcetera.

  4. 4.

${\mathrm{Mat}}^{n,m}\left({K}\right)$ denotes the set of block matrices
$A=\left({A_{i,j}}\right)_{i,j\geq 0}$ with
$A_{i,j}:{\mathrm{Sym}}^{i}K^{m}\to{\mathrm{Sym}}^{j}K^{n},$ hence
$A_{i,j}\in\mathrm{M}_{d_{n,i}\times
d_{m,j}}\left({K}\right)=\mathrm{Mat}^{i,j}_{n,m}\left({K}\right)$:

$A=\left({\begin{tabular}[]{>{\centering\arraybackslash}m{.5cm}|>{\centering\arraybackslash}m{2.2cm}|>{\centering\arraybackslash}m{1.2cm}|>{\centering\arraybackslash}m{-0.3cm}|c}$\ddots$\@add@centering&$\vdots$\@add@centering&$\vdots$\@add@centering&$\vdots$\@add@centering&\\\
\cline{1-5}\cr\hbox{\multirowsetup$\cdots$}\@add@centering&\hbox{\multirowsetup$A_{2,2}$}\@add@centering&\hbox{\multirowsetup$A_{2,1}$}\@add@centering&\hbox{\multirowsetup}\@add@centering&\hbox{\multirowsetup$\leftarrow
A_{2,0}$}\\\
\@add@centering&\@add@centering&\@add@centering&\@add@centering&\\\
\@add@centering&\@add@centering&\@add@centering&\@add@centering&\\\
\@add@centering&\@add@centering&\@add@centering&\@add@centering&\\\
\@add@centering&\@add@centering&\@add@centering&\@add@centering&\\\
\cline{1-5}\cr\hbox{\multirowsetup$\cdots$}\@add@centering&\hbox{\multirowsetup$A_{1,2}$}\@add@centering&\hbox{\multirowsetup$A_{1,1}$}\@add@centering&\hbox{\multirowsetup}\@add@centering&\hbox{\multirowsetup$\leftarrow
A_{1,0}$}\\\
\@add@centering&\@add@centering&\@add@centering&\@add@centering&\\\
\@add@centering&\@add@centering&\@add@centering&\@add@centering&\\\
\cline{1-5}\cr$\cdots$\@add@centering&$A_{0,2}$\@add@centering&$A_{0,1}$\@add@centering&\@add@centering&${\leftarrow
A_{0,0}}$\\\ \cline{1-5}\cr\end{tabular}}\right)$

We write ${\mathrm{Mat}}:={\mathrm{Mat}}^{n,n}$ if $n$ is unambiguous.
Conversely, $\mathrm{Mat}^{i,j}_{n,m}$ is embedded in ${\mathrm{Mat}}^{n,m}$
by identifying every matrix $A_{i,j}$ with an element of
${\mathrm{Mat}}^{n,m}$ equal to $0$ save for block $A_{i,j}$.

######  Definition 1.2.3 (Symmetric products of finite and infinite matrices).

[4, 34]

  1. (i)

Let $A\in{\mathrm{Mat}}_{m,n}^{i_{1},j_{1}}\left({K}\right)$,
$B\in{\mathrm{Mat}}_{m,n}^{i_{2},j_{2}}\left({K}\right)$. Given any multi-
index $\mathbf{k}=\left({k_{1},\dots,k_{n}}\right)\in{\mathbb{Z}}^{n}_{\geq
0}$ such that $\left|{\mathbf{k}}\right|=k_{1}+\dots+k_{n}=j_{1}+j_{2}$,
define $C:=A\odot B\in{\mathrm{Mat}}_{m,n}^{i_{1}+i_{2},j_{1}+j_{2}}$ by

$C\left({\mbox{\boldmath${e}$\unboldmath}_{1}^{\odot
k_{1}}\cdots\mbox{\boldmath${e}$\unboldmath}_{n}^{\odot
k_{n}}}\right)=\frac{1}{\binom{j_{1}+j_{2}}{j_{1}}}\sum_{\mathbf{{p}}}\binom{\mathbf{k}}{\mathbf{p}}A\left({\mbox{\boldmath${e}$\unboldmath}_{1}^{\odot
p_{1}}\cdots\mbox{\boldmath${e}$\unboldmath}_{n}^{\odot
p_{n}}}\right)\\!\odot\\!B\left({\mbox{\boldmath${e}$\unboldmath}_{1}^{\odot
k_{1}-p_{1}}\cdots\mbox{\boldmath${e}$\unboldmath}_{n}^{\odot
k_{n}-p_{n}}}\right),$ (5)

notation abused by removing $\odot$ to reduce space, binomials as in (4) and
summation taking place for specific multi-indices $\mathbf{p}$, namely those
such that

$\left|{\mathbf{p}}\right|=j_{1}\qquad\mbox{ and \qquad}0\leq p_{i}\leq
k_{i},\quad i=1,\dots,n.$

  2. (ii)

For any $A,B\in{\mathrm{Mat}}^{n,m}\left({K}\right)$, define $A\odot
B=C\in{\mathrm{Mat}}^{n,m}\left({K}\right)$ by

$C=\left({C_{i,j}}\right)_{i,j\geq 0},\qquad C_{i,j}=\sum_{0\leq i_{1}\leq
i,\;0\leq j_{1}\leq j}\binom{j}{j_{1}}A_{i_{1},j_{1}}\odot
B_{i-i_{1},j-j_{1}}.$ (6)

Same as always, ⊙k will stand for powers built with this product.

The following is a mere exercise in induction:

######  Lemma 1.2.4.

Defining $\bigodot_{i=1}^{r}A_{i}$ recursively by
$\left({\bigodot_{i=1}^{r-1}A_{i}}\right)\odot A_{r}$ with
$A_{i}\in{\mathrm{Mat}}_{m,n}^{k_{i},j_{i}}$,

$\left({A_{1}\odot\cdots\odot
A_{r}}\right)\mbox{\boldmath${e}$\unboldmath}^{\odot\mathbf{{k}}}=\frac{1}{\binom{j_{1}+\dots+j_{r}}{j_{1},j_{2},\dots,j_{r}}}\sum_{\mathbf{{p}}_{1},\dots,\mathbf{{p}}_{r}}\binom{\mathbf{{k}}}{\mathbf{{p}}_{1},\dots,\mathbf{{p}}_{r}}\bigodot_{i=1}^{r}A_{i}\mbox{\boldmath${e}$\unboldmath}^{\odot\mathbf{{p}}_{i}},$
(7)

if $\left|{\mathbf{{k}}}\right|=j_{1}+\dots+j_{r}$, sums obviously taken for
$\mathbf{{p}}_{1}+\dots+\mathbf{{p}}_{r}=\mathbf{{k}}$ and
$\left|{\mathbf{{p}}_{i}}\right|=j_{i}$, for every $i=1,\dots,r$. $\square$

The following is straightforward and has already been seen e.g. in [4, 5, 34],
or can be easily derived therefrom:

######  Proposition 1.2.5.

For any $A$, $B$, $C$, and whenever products make sense,

  1. a)

$A\odot B=B\odot A$.

  2. b)

$\left({A+B}\right)\odot C=A\odot C+B\odot C$.

  3. c)

$\left({A\odot B}\right)\odot C=A\odot\left({B\odot C}\right)$.

  4. d)

$\left({\alpha A}\right)\odot B=\alpha\left({A\odot B}\right)$ for every
$\alpha\in K$.

  5. e)

If $A$ is square and invertible, then $\left({A^{-1}}\right)^{\odot
k}=\left({A^{\odot k}}\right)^{-1}$.

  6. f)

$A\odot B=0$ if and only if $A=0$ or $B=0$.

  7. g)

If $A$ is a square $(1,1)$-matrix, then
$A\mbox{\boldmath${v}$\unboldmath}_{1}\odot
A\mbox{\boldmath${v}$\unboldmath}_{2}\odot\cdots\odot
A\mbox{\boldmath${v}$\unboldmath}_{m}=A^{\odot
m}\mbox{\boldmath${v}$\unboldmath}_{1}\odot\cdots\odot\mbox{\boldmath${v}$\unboldmath}_{m}.$

  8. h)

If ${v}$ is a column vector, then $\left({A\odot
B}\right)\mbox{\boldmath${v}$\unboldmath}^{\odot\left({p+q}\right)}=\left({A\mbox{\boldmath${v}$\unboldmath}^{\odot
p}}\right)\odot\left({B\mbox{\boldmath${v}$\unboldmath}^{\odot q}}\right)$,
$p,q\in{\mathbb{Z}}_{\geq 0}.$

######  Lemma 1.2.6 ([34]).

  1. (i)

Given square $A,B\in{\mathrm{Mat}}_{n}^{k,k}$ and matrices
$X_{i}\in{\mathrm{Mat}}_{n}^{k,j_{i}},\,i=1,2$,

$\left({A\odot B}\right)\left({X_{1}\odot
X_{2}}\right)=\frac{1}{2}\left({AX_{1}\odot BX_{2}+BX_{1}\odot
AX_{2}}\right),$ (8)

and in general for any square $A_{1},\dots,A_{m}\in{\mathrm{Mat}}_{n}^{k,k}$
and any $X_{i}\in{\mathrm{Mat}}_{n}^{k,j_{i}}$, $i=1,\dots,m$,

$\left({\bigodot_{i=1}^{m}A_{i}}\right)\left({\bigodot_{i=1}^{m}X_{i}}\right)=\frac{1}{m!}\sum_{\sigma\in{\mathfrak{S}}_{k}}\bigodot_{i=1}^{m}A_{\sigma\left({i}\right)}X_{i}.\qquad\square$
(9)

  2. (ii)

Given $A\in{\mathrm{Mat}}_{n}^{1,j}$ and $X_{1},\dots,X_{m}$ such that
$X_{i}\in{\mathrm{Mat}}_{n}^{1,q_{i}}$, $1\leq j\leq m$,

$\binom{m}{j}\\!\left({A\odot{\mathrm{Id}}^{\odot
m-j}_{n}}\right)\\!\bigodot_{i=1}^{m}X_{i}=\sum_{1\leq i_{1}<\dots<i_{j}\leq
m}\\!\left[{A\left({X_{i_{1}}\odot\cdots\odot
X_{i_{j}}}\right)}\right]\odot\\!\bigodot_{s\neq
i_{1},\dots,i_{j}}X_{s}.\quad\square$ (10)

  3. (iii)

Given a square matrix $A\in{\mathrm{Mat}}_{n}^{1,1}$ and $X_{1},\dots,X_{m}$
such that $X_{i}\in{\mathrm{Mat}}_{n}^{1,j_{i}}$,

$\left({A\odot{\mathrm{Id}}^{\odot
m-1}_{n}}\right)\left({\bigodot_{i=1}^{m}X_{i}}\right)=\frac{1}{m}\sum_{i=1}^{m}\left({AX_{i}}\right)\odot\left({X_{1}\odot\cdots\odot\widehat{X_{i}}\odot\cdots\odot
X_{m}}\right).$ (11)

  4. (iv)

Given square matrix $X\in{\mathrm{Mat}}_{n}^{k-1,k-1}$ and vector
$\mbox{\boldmath${v}$\unboldmath}\in K^{n}$, we have, for each $i=1,\dots,n$,

$\left({X\odot\mbox{\boldmath${e}$\unboldmath}_{i}^{T}}\right)\left({\mbox{\boldmath${v}$\unboldmath}\odot{\mathrm{Id}}_{n}^{\odot
k-1}}\right)=\frac{k-1}{k}X\left({\mbox{\boldmath${v}$\unboldmath}\odot{\mathrm{Id}}_{n}^{\odot
k-2}}\right)\odot\mbox{\boldmath${e}$\unboldmath}_{i}^{T}+\frac{v_{i}}{k}X$
(12)

######  Lemma 1.2.7.

Let $\left({K,\partial}\right)$ be a differential field.

  1. (i)

For any given $X\in{\mathrm{Mat}}_{n}^{k_{1},j_{1}}\left({K}\right)$ and
$Y\in{\mathrm{Mat}}_{n}^{k_{2},j_{2}}\left({K}\right)$,

$\partial\left({X\odot Y}\right)=\partial\left({X}\right)\odot
Y+X\odot\partial\left({Y}\right).\qquad\square$ (13)

  2. (ii)

If $Y$ is a square $n\times n$ matrix having entries in $K$ and $\partial
Y=AY$, then

$\partial\;{\mathrm{Sym}}^{k}Y=k\left({A\odot{\mathrm{Sym}}^{k-1}\left({{\mathrm{Id}}_{n}}\right)}\right){\mathrm{Sym}}^{k}Y.$
(14)

  3. (iii)

If $X\in{\mathrm{Mat}}_{n}^{1,j_{1}}$ and $Y\in{\mathrm{Mat}}_{n}^{1,j_{2}}$
satisfy systems $\partial X=AX+B_{1}$ and $\partial Y=AY+B_{2}$ with
$A\in{\mathrm{Mat}}_{n}^{1,1},B_{i}\in{\mathrm{Mat}}_{n}^{1,j_{i}},$ then
symmetric product $X\odot Y$ satisfies linear system

$\partial\left({X\odot
Y}\right)=2\left({A\odot{\mathrm{Id}}_{d_{n,k}}}\right)\left({X\odot
Y}\right)+\left({B_{1}\odot Y+B_{2}\odot X}\right).$ (15)

  4. (iv)

If $\partial X_{i}=AX_{i}+B_{i},\;i=1,\dots,m,$ with
$X_{i},B_{i}\in{\mathrm{Mat}}_{n}^{1,j_{i}}$, $A\in{\mathrm{Mat}}_{n}^{1,1}$
then

$\partial\bigodot_{i=1}^{m}X_{i}=m\left({A\odot{\mathrm{Id}}_{d_{n,k}}^{\odot
m-1}}\right)\bigodot_{i=1}^{m}X_{i}+\sum_{i=1}^{m}B_{i}\odot\bigodot_{j\neq
i}X_{j}.\qquad\square$ (16)

######  Definition 1.2.8.

(See also [5]) for every matrix $A\in{\mathrm{Mat}}^{n,m}$ we define the
formal power series

$\exp_{\odot}A:=1+A^{\odot 1}+\frac{1}{2}A^{\odot
2}+\cdots=\sum_{i=0}^{\infty}\frac{1}{i!}A^{\odot i}.$

Whenever $A=0$ save for a finite distinguished submatrix $A_{j,k}$, the abuse
of notation $\exp_{\odot}A_{j,k}=\exp_{\odot}A$ will be customary.

######  Lemma 1.2.9.

  1. hola
  2. a)

For every two $A,B\in{\mathrm{Mat}}^{n,m}$,
$\exp_{\odot}\left({A+B}\right)=\exp_{\odot}{A}\odot\exp_{\odot}{B}$.

  3. b)

For every $Y\in{\mathrm{Mat}}^{n,m}$ and any derivation $\partial:K\to K$,
$\partial\exp_{\odot}Y=\left({\partial Y}\right)\odot\exp_{\odot}Y.$

  4. c)

([4, Corollary 3]) Given square matrices $A,B\in{\mathrm{Mat}}^{1,1}_{n}$,
$\exp_{\odot}AB=\exp_{\odot}A\exp_{\odot}B$.

  5. d)

In particular, for every invertible square $A\in{\mathrm{Mat}}^{1,1}_{n}$,
$\exp_{\odot}A^{-1}=\left({\exp_{\odot}A}\right)^{-1}$. $\square$

For examples and properties, see [34, §3.1].

###  1.3 Application to power series

######  Lemma 1.3.1 ([34, Lemma 3.7]).

If $F=F_{1}\times\dots\times F_{m}$ is a _vector_ power series, adequate
$M_{F}^{1,i}\in{\mathrm{Mat}}_{m,n}^{1,i}\left({K}\right)$ render

$\framebox{$F\left({\mbox{\boldmath${x}$\unboldmath}}\right)=M_{F}\exp_{\odot}X$}\quad\mbox{
where
}M_{F}:=\left({\begin{tabular}[]{ccc|c}$\cdots$&$M_{F}^{1,2}$&$M_{F}^{1,1}$&$M_{F}^{1,0}$\\\
\hline\cr$\cdots$&$0$&$0$&$0$\end{tabular}}\right)\in{\mathrm{Mat}}^{m,n}.$

Following Definition 1.2.8, write
$F\left({\mbox{\boldmath${x}$\unboldmath}}\right)=M_{F}\exp_{\odot}\mbox{\boldmath${x}$\unboldmath}$
if it poses no clarity issue. $\square$

Thus every formal power series $F\in
K\left[{\left[{\mbox{\boldmath${x}$\unboldmath}}\right]}\right]$,
$\mbox{\boldmath${x}$\unboldmath}=\left({x_{1},\dots,x_{n}}\right)$ can be
expressed in the form $M_{F}\exp_{\odot}{\mbox{\boldmath${x}$\unboldmath}}$,
where

$M_{F}:=\left({\begin{tabular}[]{ccc|c}$\cdots$&$M_{F}^{1,2}$&$M_{F}^{1,1}$&$M_{F}^{1,0}$\\\
\hline\cr$\cdots$&$0$&$0$&$0$\end{tabular}}\right)\in{\mathrm{Mat}}^{1,n}\left({K}\right),\qquad
X:=\left({\begin{tabular}[]{c|c}$0$&$\mbox{\boldmath${x}$\unboldmath}$\\\
\hline\cr$0$&$0$\end{tabular}}\right).$

$M_{F}=J_{F}+M_{F}^{1,0}:=\left({\begin{tabular}[]{ccc|c}$\cdots$&$M_{F}^{1,2}$&$M_{F}^{1,1}$&$0$\\\
\hline\cr$\cdots$&$0$&$0$&$0$\end{tabular}}\right)+\left({\begin{tabular}[]{c|c}$0$&$M_{F}^{1,0}$\\\
\hline\cr$0$&$0$\end{tabular}}\right).$ (17)

See [34, §3.2] for more details.

###  1.4 Higher-order variational equations

See [34] for proofs and further elaboration on everything summarized below:

######  Notation 1.4.1.

For every set of indices $1\leq i_{1}\leq\dots\leq i_{r}$ such that
$\sum_{j=1}^{r}i_{j}=k$, $c^{k}_{i_{1},\dots,i_{r}}$ is defined as the amount
of totally ordered partitions of a set of $k$ elements among subsets of sizes
$i_{1},\dots,i_{r}$:

$c_{i_{1},\dots,i_{j}}=c^{k}_{i_{1},\dots,i_{j}}=\frac{\binom{k}{i_{1}\,i_{2}\,\cdots\,i_{j}}}{n_{1}!\cdots
n_{m}!},\quad\left\\{\\!\begin{array}[]{l}\left({i_{1},\dots,i_{j}}\right)=\left({k_{1}\stackrel{{\scriptstyle
n_{1}}}{{\dots}}k_{1},\cdots,k_{m}\stackrel{{\scriptstyle
n_{m}}}{{\dots}}k_{m}}\right),\\\ 1\leq
k_{1}<k_{2}<\dots<k_{m}.\end{array}\right.$ (18)

######  Proposition 1.4.2.

Let $K={\mathbb{C}}\left({\mbox{\boldmath${\phi}$\unboldmath}}\right)$ and
$A_{k}$, $Y_{k}$ be
$\partial_{k}X\left({\mbox{\boldmath${\phi}$\unboldmath}}\right)$,
$\partial^{k}_{z}\psi\left({t,\mbox{\boldmath${\phi}$\unboldmath}}\right)$
minus crossed terms; let $A=J_{X}^{\phi},Y=J_{\phi}$ be the derivative jets
for $X$ and $\psi$ at ${\phi}$, with $A_{k}$, $Y_{k}$ as blocks. Then, if
$c^{k}_{\mathbf{i}}=\\#\left\\{{\mbox{ordered $i_{1},\dots,i_{r}$-partitions
of $k$ elements}}\right\\}$,

$\dot{\overbracket{Y_{k}}}=\sum_{j=1}^{k}A_{j}\sum_{\left|{\mathbf{i}}\right|=k}c^{k}_{\mathbf{i}}{\small
Y_{i_{1}}\odot Y_{i_{2}}\odot\cdots\odot Y_{i_{j}}},\quad\mbox{$k\geq 1$}.$
(${\mathrm{VE}^{k}_{\phi}}$)

######  Lemma 1.4.3.

Let $Y\in{\mathrm{Mat}}\left({K}\right)$ equal to zero outside of block row
1,k, $k\geq 1$:

$Y:=\left({\begin{tabular}[]{cccc|c}$\cdots$&$Y_{3}$&$Y_{2}$&$Y_{1}$&$0$\\\
\hline\cr$0$&$0$&$0$&$0$&$0$\end{tabular}}\right),\qquad
Y_{i}\in{\mathrm{Mat}}^{1,i}_{n}.$ (19)

Let $Z_{r,s}$, $s,r\geq 1$, be the corresponding block in $\exp_{\odot}Y$.
Then,

  1. a)

Row block $r$ in $\exp_{\odot}Y$ is recursively obtained in terms of row
blocks $1$ and $r-1$:

$Z_{r,s}=\frac{1}{r}\sum_{j=1}^{s-r+1}\binom{s}{j}Y_{j}\odot Z_{r-1,s-j}.$
(20)

In particular, $Z_{r,r}=Y_{1}^{\odot r}$ and $Z_{r,s}=0_{d_{n,r},d_{n,s}}$
whenever $r>s$.

  2. b)

Using Notation 1.4.1 and (18), for every $s\geq r$

$Z_{r,s}=\sum_{i_{1}+\dots+i_{r}=s}c^{s}_{i_{1},\dots,i_{r}}Y_{i_{1}}\odot
Y_{i_{2}}\odot\cdots\odot Y_{i_{r}}.$ (21)

######  Notation 1.4.4.

$K:={\mathbb{C}}\left({\mbox{\boldmath${\phi}$\unboldmath}}\right)$,
$A_{i}:=X^{\left({i}\right)}\\!\left({\mbox{\boldmath${\phi}$\unboldmath}}\right)$,
$Y_{i}:=\mathrm{lex}\left({\frac{\partial^{i}}{\partial\mbox{\boldmath${z}$\unboldmath}^{i}}\varphi{\left({t,\mbox{\boldmath${\phi}$\unboldmath}}\right)}}\right)$
and, per Lemma 1.4.3,

$\Upsilon_{1}=Y_{1},\qquad\Upsilon_{k}=\left({\begin{tabular}[]{ccc}$Z_{k,k}$&\\\
\cline{2-3}\cr$Z_{k-1,k}$&\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol\\\
\vdots&\vrule\lx@intercol\hfil$\phantom{a}\Upsilon_{k-1}\phantom{a}$\hfil\lx@intercol\vrule\lx@intercol
\\\ $Z_{1,k}$&\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{2-3}\cr\end{tabular}}\right),\quad k\geq 2,$ (22)

formed by the first $k$ block rows and columns in $\Upsilon=\exp_{\odot}Y$.
Define $A,Y\in{\mathrm{Mat}}\left({K}\right)$ as in Lemma 1.4.3 with the above
$A_{i}$, $Y_{i}$ as blocks. Denote the canonical basis on $K^{n}$ (meaning the
set of columns of ${\mathrm{Id}}_{n}$) by
$\left\\{{\mbox{\boldmath${e}$\unboldmath}_{1},\dots,\mbox{\boldmath${e}$\unboldmath}_{n}}\right\\}$.

######  Proposition 1.4.5 (Explicit version of $\mathrm{LVE}_{\phi}^{k}$).

Still following Notation 1.4.4, the infinite system

$\framebox{$\dot{X}=A_{\mathrm{LVE}_{\phi}}X,$}\qquad\qquad
A_{\mathrm{LVE}_{\phi}}:=A\odot\exp_{\odot}{\mathrm{Id}}_{n},$
($\mathrm{LVE}_{\phi}$)

has $\Upsilon:=\exp_{\odot}Y$ as a solution matrix. Hence, for every $k\geq
1$,

  1. a)

the lower-triangular recursive $D_{n,k}\times D_{n,k}$ form for
$\mathrm{LVE}_{\phi}^{k}$ is $\dot{Y}=A_{\mathrm{LVE}_{\phi}^{k}}Y$, its
system matrix being obtained from the first $k$ row and column blocks of
$A_{\mathrm{LVE}_{\phi}}$:

$A_{\mathrm{LVE}_{\phi}^{k}}=\left({\begin{tabular}[]{ccc}$\binom{k}{k-1}A_{1}\odot{\mathrm{Id}}_{n}^{\odot{k-1}}$&\\\
\cline{2-3}\cr$\binom{k}{k-2}A_{2}\odot{\mathrm{Id}}_{n}^{\odot
k-2}$&\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol\\\
\vdots&\vrule\lx@intercol\hfil$A_{\mathrm{LVE}_{\phi}^{k-1}}$\hfil\lx@intercol\vrule\lx@intercol
\\\
$\binom{k}{0}A_{k}$&\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol
\\\ \cline{2-3}\cr\end{tabular}}\right),$ (23)

  2. b)

and the principal fundamental matrix for $\mathrm{LVE}_{\phi}^{k}$ is
$\Upsilon_{k}$ from $\exp_{\odot}Y$ in Notation 1.4.4.

The construction of an infinite matrix
$\Upsilon_{\mathrm{LVE}_{\phi}}=\exp_{\odot}Y$ follows in such a way, that the
first $d_{n,k}:=\binom{n+k-1}{n-1}$ rows and columns are a principal
fundamental matrix for $\mathrm{LVE}_{\phi}^{k}$. The definition is recursive
and amenable to symbolic computation.

######  Example 1.4.6.

For instance, for $k=5$ we have

$A_{\mathrm{LVE}_{\phi}^{5}}=\left({\begin{array}[]{ccccc}5A_{1}\odot{\mathrm{Id}}_{n}^{\odot
4}&&&&\\\ 10A_{2}\odot{\mathrm{Id}}_{n}^{\odot
3}&4A_{1}\odot{\mathrm{Id}}_{n}^{\odot 3}&&&\\\
10A_{3}\odot{\mathrm{Id}}_{n}^{\odot 2}&6A_{2}\odot{\mathrm{Id}}_{n}^{\odot
2}&3A_{1}\odot{\mathrm{Id}}_{n}^{\odot 2}&&\\\
5A_{4}\odot{\mathrm{Id}}_{n}&4A_{3}\odot{\mathrm{Id}}_{n}&3A_{2}\odot{\mathrm{Id}}_{n}&2A_{1}\odot{\mathrm{Id}}_{n}&\\\
A_{5}&A_{4}&A_{3}&A_{2}&A_{1}\end{array}}\right),$

and the principal fundamental matrix $\Upsilon_{5}$ is

$\left({\begin{array}[]{ccccc}Y_{1}^{\odot 5}&&&&\\\ 10Y_{1}^{\odot 3}\odot
Y_{2}&Y_{1}^{\odot 4}&&&\\\ 10Y_{1}^{\odot 2}\odot Y_{3}+15Y_{1}\odot
Y_{2}^{\odot 2}&6Y_{1}^{\odot 2}\odot Y_{2}&Y_{1}^{\odot 3}&&\\\ 10Y_{2}\odot
Y_{3}+5Y_{1}\odot Y_{4}&4Y_{1}\odot Y_{3}+3Y_{2}\odot Y_{2}&3Y_{1}\odot
Y_{2}&Y_{1}^{\odot 2}\\\ Y_{5}&Y_{4}&Y_{3}&Y_{2}&Y_{1}\end{array}}\right),$
(24)

hence (${\mathrm{VE}^{k}_{\phi}}$) for $k=5$ is the lowest row in
$A_{\mathrm{LVE}_{\phi}^{5}}$ times the leftmost column in $\Upsilon_{5}$.

###  1.5 First integrals and higher-order variational equations

Let $F:U\subseteq{\mathbb{C}}^{n}\to{\mathbb{C}}^{n}$ be a holomorphic
function and $\mbox{\boldmath${\phi}$\unboldmath}:I\subset{\mathbb{C}}\to U$.
Firstly, the flow $\varphi\left({t,\mbox{\boldmath${z}$\unboldmath}}\right)$
of $X$ admits, at least formally, Taylor expansion (2) along ${\phi}$ which is
expressible as

$\varphi\left({t,\mbox{\boldmath${\phi}$\unboldmath}+\mbox{\boldmath${\xi}$\unboldmath}}\right)=\mbox{\boldmath${\phi}$\unboldmath}+Y_{1}\mbox{\boldmath${\xi}$\unboldmath}+\frac{1}{2}Y_{2}\mbox{\boldmath${\xi}$\unboldmath}^{\odot
2}+\dots=\mbox{\boldmath${\phi}$\unboldmath}+J_{\phi}\exp_{\odot}\mbox{\boldmath${\xi}$\unboldmath},$
(25)

where $J_{\phi}$ is the jet for flow $\varphi\left({t,\cdot}\right)$ along
${\phi}$, displayed as $Y$ in (19) and defined in Notation 1.4.4 – that is,
the matrix whose $\odot$-exponential $\Upsilon$ is a solution matrix for
($\mathrm{LVE}_{\phi}$). Secondly, the Taylor series of $F$ along ${\phi}$ can
be written, cfr. [3, Lemma 2] and Notation 1.2.1,

$F\left({\mbox{\boldmath${y}$\unboldmath}+\mbox{\boldmath${\phi}$\unboldmath}}\right)=F\left(\mbox{\boldmath${\phi}$\unboldmath}\right)+\sum^{\infty}_{m=1}\frac{1}{m!}\left\langle
F^{(m)}\left(\mbox{\boldmath${\phi}$\unboldmath}\right)\,,\,{\mathrm{Sym}}^{m}\mbox{\boldmath${y}$\unboldmath}\right\rangle.$
(26)

Lemma 1.3.1 and (17) trivially implies (26) can be expressed as
$F\left({\mbox{\boldmath${y}$\unboldmath}+\mbox{\boldmath${\phi}$\unboldmath}}\right)=M^{\phi}_{F}\exp_{\odot}\mbox{\boldmath${y}$\unboldmath}$,
where

$M^{\phi}_{F}=J^{\phi}_{F}+F^{\left({0}\right)}\\!\left({\mbox{\boldmath${\phi}$\unboldmath}}\right):=\left({\begin{tabular}[]{ccc|c}$\cdots$&$0$&$0$&$0$\\\
$\cdots$&$F^{\left({2}\right)}\\!\left({\mbox{\boldmath${\phi}$\unboldmath}}\right)$&$F^{\left({1}\right)}\\!\left({\mbox{\boldmath${\phi}$\unboldmath}}\right)$&$F^{\left({0}\right)}\\!\left({\mbox{\boldmath${\phi}$\unboldmath}}\right)$\\\
\hline\cr$\cdots$&$0$&$0$&$0$\end{tabular}}\right)\in{\mathrm{Mat}}^{1,n}\left({K}\right),$

i.e. $J_{F}^{\phi}$ is the jet or horizontal strip of lex-sifted partial
derivatives of $F$ at ${\phi}$.

Let

$\framebox{$\dot{X}=A_{\mathrm{LVE}^{\star}_{\phi}}X,$}\qquad\qquad
A_{\mathrm{LVE}_{\phi}^{\star}}:=-\left({A\odot\exp_{\odot}{\mathrm{Id}}_{n}}\right)^{T},$
($\mathrm{LVE}_{\phi}^{\star}$)

be the adjoint or dual variational system of (1) along $\phi$.

It is immediate, upon derivation of equation
$\Upsilon_{k}\Upsilon_{k}^{-1}={\mathrm{Id}}_{D_{n,k}}$, that

######  Lemma 1.5.1.

$\left({\Upsilon^{-1}_{k}}\right)^{T}$ is a principal fundamental matrix of
$\left({\mathrm{LVE}_{\phi}^{k}}\right)^{\star}$, $k\geq 1$, hence
$\lim_{k}\left({\Upsilon_{k}^{-1}}\right)^{T}$, is a solution to
($\mathrm{LVE}_{\phi}^{\star}$).

The following was proven in [30] and recounted in [3, Lemma 7], and was may
now be expressed in a simple, compact fashion:

######  Lemma 1.5.2.

Let $F$ and ${\phi}$ be a holomorphic first integral and a non-constant
solution of (1) respectively. Let $V:=J_{F}^{T}$ be the transposed jet of $F$
along ${\phi}$. Then, $V$ is a solution of ($\mathrm{LVE}_{\phi}^{\star}$).

Hence the issue, already considered in [3], is whether some converse holds and
a solution of the dual system $\left(\mathrm{LVE}^{k}_{\phi}\right)^{\star}$
is or can be the set of first $k$ terms of the formal (or perhaps even
convergent) series form of a first integral. This was addressed in the
aforementioned reference and was put in compact, explicit linearized form in
[34], and will be recounted below.

Define

$\widehat{A}:=\left({\begin{tabular}[]{ccc|c}$\cdots$&$0$&$0$&$0$\\\
$\cdots$&$A_{2}$&$A_{1}$&$A_{0}$\\\
\hline\cr$\cdots$&$0$&$0$&$0$\end{tabular}}\right),\qquad
A_{i}:=X^{\left({i}\right)}\\!\left({\mbox{\boldmath${\phi}$\unboldmath}}\right)\in{\mathrm{Mat}}^{1,i}_{n}\left({K}\right),\qquad
A_{0}:=X\left({\mbox{\boldmath${\phi}$\unboldmath}}\right)=\dot{\mbox{\boldmath${\phi}$\unboldmath}},$
(27)

and let

$\widehat{A}_{\mathrm{LVE}_{\phi}}:=\widehat{A}\odot\exp_{\odot}{\mathrm{Id}}_{n}=\lim_{k}\widehat{A}_{\mathrm{LVE}^{k}_{\phi}}=\lim_{k}\left({\begin{tabular}[]{|c|c}\cline{1-1}\cr$\binom{k}{k}X^{\left({0}\right)}\left({\mbox{\boldmath${\phi}$\unboldmath}}\right)\odot{\mathrm{Id}}_{n}^{\odot
k}$&\\\
\cline{1-2}\cr$\binom{k}{k-1}X^{\left({1}\right)}\left({\mbox{\boldmath${\phi}$\unboldmath}}\right)\odot{\mathrm{Id}}_{n}^{\odot
k-1}$&\lx@intercol\hfil\hbox{\multirowsetup$\widehat{A}_{\mathrm{LVE}^{k-1}_{\phi}}$}\hfil\lx@intercol\vrule\lx@intercol
\\\
\cline{1-1}\cr$\binom{k}{k-2}X^{\left({2}\right)}\left({\mbox{\boldmath${\phi}$\unboldmath}}\right)\odot{\mathrm{Id}}_{n}^{\odot
k-2}$&\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr$\vdots$&\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol
\\\
\cline{1-1}\cr$\binom{k}{0}X^{\left({k}\right)}\left({\mbox{\boldmath${\phi}$\unboldmath}}\right)\odot{\mathrm{Id}}_{n}^{\odot
0}$&\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol \\\
\hline\cr\end{tabular}}\right),$

The following gave a compact form to [3, Th. 12] in terms of $\odot$ and
infinite matrices:

######  Proposition 1.5.3 ([34, Prop. 6]).

Let $F$, ${\phi}$, $V=\left({\cdots\mid V_{3}\mid V_{2}\mid V_{1}}\right)$ as
in Lemma 1.5.2. Then $\widehat{A}_{\mathrm{LVE}_{\phi}}^{T}V=0$. More
specifically,
$\widehat{A}_{\mathrm{LVE}_{\phi}^{k-1}}\left({V_{k}\mid\cdots\mid V_{2}\mid
V_{1}}\right)=0$ for every $k\geq 1$, i.e.

$\sum_{j=0}^{k-1}\binom{k-1}{j}\left({A_{j}\odot{\mathrm{Id}}_{n}^{k-1-j}}\right)^{T}V_{k-j}=0,\qquad\mbox{for
every }k\geq 1.\quad\square$ (28)

Hence, blocks in
$V_{1},\left({V_{2},V_{1}}\right)^{T},\left({V_{3},V_{2},V_{1}}\right)^{T},\dots$
having all entries in the base field $K$ and satisfying both equations in
Proposition 1.5.3 and 1.5.2 are jet blocks
$F^{\left({1}\right)},F^{\left({2}\right)},\dots$ of a formal series that will
be a first integral if convergent. These blocks belonging to the intersection
of $\ker\widehat{A}_{\mathrm{LVE}^{k-1}_{\phi}}^{T}$ and the solution subspace
$\mathrm{Sol}_{K}\left({\mathrm{LVE}^{k}_{\phi}}\right)^{\star}$ were called
_admissible_ solutions of the order-$k$ adjoint system in [3]. We will call
the sum constructed from an admissible solution a formal first integral.

For a short introduction on how changes of variables are reflected on this
scheme, as well as the explicit form for the _monodromy matrix_ [39] of
${\mathrm{LVE}_{\phi}^{k}}$ along any path $\gamma$, see of [34, §4 & 5].

##  2 Jet filtering methods

###  2.1 Preliminaries

Let $f$ be a first integral of (1). The following are immediate:

######  Lemma 2.1.1.

  1. (i)

For every $i\geq 1$,

$\left({fg}\right)^{\left({i}\right)}=\sum_{j=0}^{i}\binom{i}{j}f^{\left({j}\right)}\odot
g^{\left({i-j}\right)},$ (29)

with the understanding that symmetric product by a scalar is identical to the
usual product, hence

$\displaystyle J\left\\{{fg}\right\\}$ $\displaystyle=$ $\displaystyle
U\odot\left({U^{\odot 2}}\right)^{T}\left[{J\left\\{{f}\right\\}\odot
J\left\\{{g}\right\\}}\right]=U\odot\left({U^{\odot
2}}\right)^{T}\left[{\exp_{\odot}J\left\\{{f}\right\\}\odot\exp_{\odot}J\left\\{{g}\right\\}}\right]$
$\displaystyle=$ $\displaystyle
U\odot\left[{U^{T}\exp_{\odot}J\left\\{{f}\right\\}}\right]\odot\left[{U^{T}\exp_{\odot}J\left\\{{g}\right\\}}\right]$

where $U=\left({\begin{tabular}[]{c|c}$0$&$1$\\\
\hline\cr$0$&$0$\end{tabular}}\right)$

  2. (ii)

For every $k\geq 1$,

$\left({f^{k}}\right)^{\left({i}\right)}=k!\sum_{j=1}^{k}\frac{f^{j-1}}{\left({j-1}\right)!}\sum_{i_{1}+\dots+i_{k-j+1}=i}c^{i}_{i_{1},\dots,i_{k-j+1}}\bigodot_{u=1}^{k-j+1}f^{\left({i_{u}}\right)},$
(30)

hence defining

$E_{1}=\left({\begin{tabular}[]{c|c}$\mbox{\boldmath${e}$\unboldmath}_{1}^{T}$&$0$\\\
\hline\cr$0$&$0$\end{tabular}}\right),\qquad
U=\left({\begin{tabular}[]{c|c}$0$&$1$\\\
\hline\cr$0$&$0$\end{tabular}}\right),$

we have

$\displaystyle J\left\\{{f^{k}}\right\\}$ $\displaystyle=$ $\displaystyle
k!\left\\{{U\odot\left({U^{\odot
k}}\right)^{T}\left[{\exp_{\odot}J\left\\{{f}\right\\}}\right]}\right\\}=k!\left\\{{U\odot\left({U^{\odot
k}}\right)^{T}\left[{\exp_{\odot}J\left\\{{f}\right\\}}\right]}\right\\}$ (31)
$\displaystyle=$ $\displaystyle
U\odot\left[{U^{T}\exp_{\odot}J\left\\{{f}\right\\}}\right]^{\odot
k}=U\odot\left[{U^{T}J\left\\{{f}\right\\}}\right]^{\odot k}$ (32)

  3. (iii)

For every $k\geq 1$,

$f^{k}\left({1/f^{k}}\right)^{\left({i}\right)}=k\sum_{j=0}^{i}\frac{\left({k+j-1}\right)!}{k!}f^{-j}\left({-1}\right)^{j}\sum_{i_{1}+\dots+i_{j}=i}c^{i}_{i_{1},\dots,i_{j}}\bigodot_{u=1}^{j}f^{\left({i_{u}}\right)},$
(33)

hence

$\displaystyle J\left\\{{1/f}\right\\}$ $\displaystyle=$
$\displaystyle\frac{1}{f}U\odot\left\\{{\left({\begin{tabular}[]{c|c}$\vdots$&$\vdots$\\\
$0$&$3!\left({-1/f}\right)^{3}$\\\ $0$&$2!\left({-1/f}\right)^{2}$\\\
$0$&$1!\left({-1/f}\right)$\\\
\hline\cr$0$&$1$\end{tabular}}\right)^{T}\exp_{\odot}\left({J\left\\{{f}\right\\}-fU}\right)}\right\\}$
(39) $\displaystyle=$ $\displaystyle
U\odot\left({\frac{1}{f}U^{T}J\left\\{{f}\right\\}}\right)^{\odot-1}=U\odot\left({\sum_{i=0}^{\infty}\left({-1}\right)^{i}\left[{\frac{1}{f}U^{T}J\left\\{{f}\right\\}-\mathbf{1}}\right]^{\odot
i}}\right)$ (40)

###  2.2 Progressive filtering

Let us describe a method to start with a jet of valuation one and then proceed
to compute each jet term of that particular function immediately in terms of
the previous ones.

######  Lemma 2.2.1.

Define

$F_{1}:=F_{i,1}:={\mathrm{Id}}_{n}-\frac{1}{X_{i}^{0}}\left({X^{0}\odot\mbox{\boldmath${e}$\unboldmath}_{i}^{T}}\right).$
(41)

The $n-1$ non-zero rows of $F_{1}Y_{1}^{-1}$ are linearly independent
admissible solutions of degree one.

###### Proof.

Let
$F_{i,1}:={\mathrm{Id}}_{n}-\frac{1}{X_{i}\left({\phi\left({0}\right)}\right)}\left({X\left({\phi\left({0}\right)}\right)\odot\mbox{\boldmath${e}$\unboldmath}_{i}^{T}}\right)$.
We have
$\widehat{A}_{\mathrm{LVE}_{\phi}^{0}}=A_{0}^{T}=X\left({\phi}\right)^{T}$.
And $\left({\Upsilon_{1}^{-1}}\right)^{T}=\left({Y_{1}^{-1}}\right)^{T}$, the
fundamental matrix for $\mathrm{VE}^{\star}_{1}$. Thus checking that (28)
holds equates to checking whether
$A_{0}^{T}\left({Y_{1}^{-1}}\right)F_{i,1}^{T}=\mbox{\boldmath${0}$\unboldmath}^{T}_{n}$,
i.e. that $F_{i,1}Y_{1}^{-1}A_{0}$ equals a column of zeros. This is easy to
prove by noticing that the basic chain rule and (1) entails
$\dot{A}_{0}=A_{1}A_{0}$ on one hand, and on the other Lemma 1.5.1 for $k=1$
implies $\frac{d}{dt}Y^{-1}=-Y^{-1}A_{1}$. Both of these facts put together
imply

$\frac{d}{dt}\left({Y_{1}^{-1}A_{0}}\right)=\dot{\overbracket{Y_{1}^{-1}}}X+Y^{-1}\dot{\overbracket{X}}=-Y^{-1}A_{1}X+Y^{-1}A_{1}X=0$

and thus
$\frac{d}{dt}\left({F_{i,1}Y_{1}^{-1}A_{0}}\right)=F_{i,1}\frac{d}{dt}\left({Y_{1}^{-1}A_{0}}\right)=0$
as well, which means $F_{i,1}Y_{1}^{-1}A_{0}$ is a constant vector. Setting
$t=t_{0}$ and using the fact that $Y_{1}$ is a principal fundamental matrix
(i.e. $Y_{1}\left({t_{0}}\right)={\mathrm{Id}}$) means that for $t=t_{0}$,
abridging notation
$X^{0}=\left({X_{1}^{0},\dots,X_{n}^{0}}\right)=X\left({\phi\left({t_{0}}\right)}\right)$,

$\displaystyle F_{i,1}Y_{1}\left({t_{0}}\right)^{-1}A_{0}$ $\displaystyle=$
$\displaystyle
F_{i,1}A_{0}=X\left({\phi\left({t_{0}}\right)}\right)-\frac{1}{X_{i}\left({\phi\left({t_{0}}\right)}\right)}\left({X\left({\phi\left({t_{0}}\right)}\right)\odot\mbox{\boldmath${e}$\unboldmath}_{i}^{T}}\right)X\left({\phi\left({t_{0}}\right)}\right)$
$\displaystyle=$ $\displaystyle\left({\begin{array}[]{c}X_{1}^{0}\\\
X_{2}^{0}\\\ \vdots\\\ X_{i}^{0}\\\ \vdots\\\ X_{n-1}^{0}\\\
X_{n}^{0}\end{array}}\right)-\frac{1}{X_{i}^{0}}\left({\begin{array}[]{ccc}\cline{1-1}\cr\cline{3-3}\cr\vrule\lx@intercol\hfil\hbox{\multirowsetup$0_{n\times\left({i-1}\right)}$}\hfil\lx@intercol\vrule\lx@intercol&X_{1}^{0}&\vrule\lx@intercol\hfil\hbox{\multirowsetup$0_{n\times\left({n-i}\right)}$}\hfil\lx@intercol\vrule\lx@intercol\\\
\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol&X_{2}^{0}&\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol\\\
\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol&\vdots&\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol\\\
\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol&X_{i}^{0}&\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol\\\
\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol&\vdots&\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol\\\
\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol&X_{n-1}^{0}&\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol\\\
\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol&X_{n}^{0}&\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol\\\
\cline{1-1}\cr\cline{3-3}\cr\end{array}}\right)\left({\begin{array}[]{c}X_{1}^{0}\\\
X_{2}^{0}\\\ \vdots\\\ X_{i}^{0}\\\ \vdots\\\ X_{n-1}^{0}\\\
X_{n}^{0}\end{array}}\right)=\mbox{\boldmath${0}$\unboldmath}_{n}.$

Still for $k=1$, we need to check that the $n-1$ non-zero columns of
$\left({F_{1}Y^{-1}}\right)^{T}$ are linearly independent. This is a
consequence of the fact that

$F_{1}^{T}=\left({\begin{array}[]{ccccccc}\cline{1-3}\cr\vrule\lx@intercol\hfil\hbox{\multirowsetup${\mathrm{Id}}_{i-1}$}\hfil\lx@intercol\vrule\lx@intercol&0&&&\\\
\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol&\vdots&&&\\\
\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol&0&&&\\\
\cline{1-3}\cr-\frac{X_{1}}{X_{i}}&\cdots&-\frac{X_{i-1}}{X_{i}}&0&-\frac{X_{i+1}}{X_{i}}&\cdots&-\frac{X_{n}}{X_{i}}\\\
\cline{5-7}\cr&&&0&\vrule\lx@intercol\hfil\hbox{\multirowsetup${\mathrm{Id}}_{n-i}$}\hfil\lx@intercol\vrule\lx@intercol\\\
&&&\vdots&\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol\\\
&&&0&\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol\\\
\cline{5-7}\cr\end{array}}\right)$

thus in $\left({Y^{-1}}\right)^{T}F_{1}^{T}$ column $i$ equals
$\mbox{\boldmath${0}$\unboldmath}_{n}$ and its remaining $n-1$ columns are the
output of subtracting multiples of column $i$ in $\left({Y^{-1}}\right)^{T}$
from its remaining (already independent) $n-1$ columns. ∎

Let $\phi\left({t}\right)$ be a particular solution of (1) whose parametric
expression can be represented with one differential equation involving one
coordinate function $z_{i}$ (case in point: (86), (99) in §3).

######  Theorem 2.2.2.

In the above hypotheses, let $f_{1}$ be one of the columns of
$\left({Y_{1}^{-1}}\right)F_{1}^{T}$. Define $X_{k}:=\left({Y_{1}^{\odot
k}}\right)^{T}$. Then each term generated by the following recursion

$f_{k}:=-X_{k}^{-1}\int
X_{k}\left\\{{\sum_{j=2}^{k}\binom{k}{j}\left({A_{j}\odot{\mathrm{Id}}_{n}^{\odot
k-j}}\right)^{T}f_{k-j+1}}\right\\}dx,$ (43)

is the degree-$k$ term of the expansion of a formal first integral

$f\left({\mbox{\boldmath${z}$\unboldmath}}\right)=f_{1}\mbox{\boldmath${z}$\unboldmath}+\frac{1}{2}f_{2}\mbox{\boldmath${z}$\unboldmath}^{\odot
2}+\frac{1}{3!}f_{3}\mbox{\boldmath${z}$\unboldmath}^{\odot 3}+\dots,$

provided that each new integral in (2.2.2) is computed by taking independent
variable $x=z_{i}\left({t}\right)$, and the limits or constants of integration
at each stage can be taken so as to ensure that

$f_{k}\left({A_{0}\odot{\mathrm{Id}}_{n}^{\odot
k-1}}\right)+\binom{k-1}{1}f_{k-1}\left({A_{1}\odot{\mathrm{Id}}^{\odot
k-2}_{n}}\right)+\dots+\binom{k-1}{k-2}f_{2}\left({A_{k-2}\odot{\mathrm{Id}}_{n}}\right)+f_{1}A_{k-1}=0.$
(44)

###### Proof.

Most of the work has been done already by Proposition 1.5.3, Lemma 2.2.1 and
the properties intrinsic to $\odot$. Firstly,
$X^{-1}_{k}=\left[{\left({Y_{1}^{-1}}\right)^{\odot k}}\right]^{T}$. Secondly,
condition (44) is nothing but a repetition of (28). In order to check the
consequences of defining (43), all it takes is to realize that the first
$d_{n,k}\times d_{n,k}$ equations of ($\mathrm{LVE}_{\phi}^{\star}$) read

$\dot{\overbracket{\left[{\left({Y_{1}^{-1}}\right)^{\odot
k}}\right]^{T}}}=-\left({A_{1}\odot{\mathrm{Id}}_{n}^{\odot
k-1}}\right)^{T}\left[{\left({Y_{1}^{-1}}\right)^{\odot k}}\right]^{T},$

which means $\left({Y_{1}^{\odot k}}\right)^{T}$ is a principal fundamental
matrix thereof; the homogeneous part of the equations satisfied by $f_{k}$
displays the same matrix, thus (43) is just plain variation of constants
applied to what is said in Lemma 1.5.2. ∎

###  2.3 Filtering with a single infinite matrix product

The above result provides a recursive method to compute formal first integrals
of arbitrary autonomous systems, thereby summing up the aims of the current
paper. The question arises, however, as to whether a _single_ infinite filter
matrix $\Phi=\exp_{\odot}F$ exists, amenable to recursive computation but
expressible in compact form and _computed without any further quadratures_ ,
which multiplied times the fundamental matrix of the dual systems (whose terms
are also computable recursively, as seen above) yields an already-trimmed
matrix comprised of linearly independent admissible solutions in a single
swipe.

Building such an infinite quadrature-free filter matrix beyond $k=1$ has a
value that is more theoretical than practical in nature, since the
computational inviability of ever-growing matrices and the effort of
discarding the aforementioned cross-products
$g^{\left({i_{1}}\right)}_{j_{1}}\odot\cdots\odot
g^{\left({i_{m}}\right)}_{j_{m}}$ get in the way of approximating jets
properly. So the question arises: why consider one such matrix? The answer
resides in the recent trends in categorification of dynamical systems (e.g.
[7]) and the potential to transport the techniques known in reductions of flat
meromorphic connections to arbitrary dynamical systems.

A considerable wealth of evidence shows one such quadrature-free matrix
exists, at least for systems defined by analytical vector fields, but a deeper
study would warrant techniques beyond the purview and goals of this article
and will thus be left for future work:

######  Conjecture 2.3.1.

Let $\phi\left({t}\right)$ be a non-trivial solution for (1) and $t_{0}$ such
that $X_{i}\left({\phi\left({t_{0}}\right)}\right)\neq 0$ for some $i$. Let
$\left({\Upsilon_{k}^{-1}}\right)^{T}$ be the principal fundamental matrix for
$\left({\mathrm{LVE}_{\phi}^{k}}\right)^{\star}$ equalling ${\mathrm{Id}}$ at
$t=t_{0}$. Use superscript 0 to denote values at $t=t_{0}$ and define
$F_{1}:=F_{i,1}$ as in (41). Then, for every $k\geq 1$, the constant matrix
$\Phi^{T}_{k}$ defined by the transpose lower left $D_{n,k}$-block of
$\Phi=\exp_{\odot}F$, where $F=\left({\cdots\mid F_{3}\mid F_{2}\mid
F_{1}}\right)$ and

$\displaystyle F_{k}$ $\displaystyle=$
$\displaystyle-\frac{1}{X^{0}_{i}}\left[{\sum_{j=0}^{k-2}\binom{k-1}{j}F_{j+1}\left({A_{k-j-1}\odot{\mathrm{Id}}^{\odot
j}}\right)}\right]U_{k},$ (45)

where

$U_{k}=\left[{\sum_{j=0}^{k-1}\binom{k}{j+1}\left({-1}\right)^{j}\left({{\mathrm{Id}}-F_{1}}\right)^{\odot
j}\odot{\mathrm{Id}}^{\odot
k-1-j}}\right]\odot\mbox{\boldmath${e}$\unboldmath}_{i}^{T}=\left[{\bigodot_{j=1}^{k-1}\left({{\mathrm{Id}}-\zeta_{k}^{j}F_{1}}\right)}\right]\odot\mbox{\boldmath${e}$\unboldmath}_{i}^{T}$
(46)

and $\zeta_{k}=\exp\left({2\pi{\mathrm{i}}/k}\right)$ is the root of unity, is
such that the last $n-1$ non-zero columns of
$\left({\Upsilon_{k}^{-1}}\right)^{T}\left({\Phi_{k}}\right)^{T}$ are linearly
independent admissible solutions
$g_{1}^{\left({\cdot}\right)},g_{2}^{\left({\cdot}\right)},\dots,g_{n-1}^{\left({\cdot}\right)}\in\mathrm{Sol}\left({\mathrm{LVE}_{\phi}^{k}}\right)^{\star}\cap\ker\widehat{A}^{T}_{\mathrm{LVE}^{k-1}_{\phi}}$
and jets of functionally independent formal first integrals
$g_{1},\dots,g_{n-1}$ of valuation $1$.

######  Remarks 2.3.2.

  1.   2. 1.

We already know all the columns of
$\left({\Upsilon_{k}^{-1}}\right)^{T}\left({\Phi_{k}}\right)^{T}$ belong to
$\mathrm{Sol}\left({\mathrm{LVE}_{\phi}^{k}}\right)^{\star}$, because
$\frac{d}{dt}\left({\Upsilon^{-1}}\right)^{T}=A_{\mathrm{LVE}^{\star}}\left({\Upsilon^{-1}}\right)^{T}$
where $\Upsilon=\exp_{\odot}Y$ , and the above finite-order products are
nothing but truncations of the right-product of
$\left({\Upsilon^{-1}}\right)^{T}$ by a constant matrix. Thus it only remains
to show that these truncations are also annihilated by left-multiplication by
$\widehat{A}^{T}_{\mathrm{LVE}^{k-1}_{\phi}}$, and that the non-zero columns
in $\left({Y_{1}^{-1}}\right)^{T}F^{T}_{1}$ are linearly independent;
independence of the full non-zero columns in higher orders will be a trivial
consequence of the latter fact.

  3. 2.

In light of Remark 1., Lemma 2.2.1 basically entails that the entire
conjecture holds true at level $k=1$ and, as seen in §2.2, that was all we
needed to compute independent jets of formal first integrals, regardless
whether 2.3.1 loses its conjectural status:

######  Proposition 2.3.3.

The conjecture is true at $t_{0}$ for every $k\geq 1$.

###### Proof.

All occurrences of $A_{i}$ from here onward will refer to constant matrices
$A_{i}^{0}=X^{\left({i}\right)}\left({\phi\left({t_{0}}\right)}\right)$. Case
$k=1$ has already been tackled paragraphs earlier; in general, we need to
prove that

$F_{k}\left({A_{0}\odot{\mathrm{Id}}^{\odot
k-1}}\right)+\binom{k-1}{1}F_{k-1}\left({A_{1}\odot{\mathrm{Id}}^{k-2}}\right)+\dots+\binom{k-1}{k-2}F_{2}\left({A_{k-2}\odot{\mathrm{Id}}}\right)+F_{1}\left({A_{k-1}}\right)=0,$
(47)

for every $k\geq 1$.

First of all, bilinearity from Prop. 1.2.5 and a simple induction argument
imply, for every set of $n\times n$ matrices $M_{1},\dots,M_{m}$,

$\bigodot_{i=1}^{m}\left({{\mathrm{Id}}+M_{i}}\right)={\mathrm{Id}}^{\odot
m}_{n}+\sum_{j=1}^{m}{\mathrm{Id}}_{n}^{\odot m-1}\odot M_{j}+\sum_{1\leq
j_{1}<j_{2}\leq m}{\mathrm{Id}}_{n}^{\odot m-2}\odot M_{j_{1}}\odot
M_{j_{2}}+\dots+M_{1}\odot\cdots\odot M_{m}$

and thus setting $m=k-1$ and $M_{i}=-\zeta^{i}_{k}F_{1}$ very basic properties
of cyclotomic polynomials imply

$U_{k}=\left[{\bigodot_{i=1}^{k-1}\left({{\mathrm{Id}}-\zeta_{k}^{i}F_{i}}\right)}\right]\odot\mbox{\boldmath${e}$\unboldmath}_{i}^{T}=\left({\sum_{j=0}^{k-1}{\mathrm{Id}}_{n}^{\odot
j}\odot F_{1}^{\odot
k-1-j}}\right)\odot\mbox{\boldmath${e}$\unboldmath}_{i}^{T}$ (48)

We claim that

$U_{k}\left({A_{0}\odot{\mathrm{Id}}_{n}^{\odot
k-1}}\right)=X_{i}{\mathrm{Id}}_{n}^{\odot k-1},\qquad\mbox{ for every }k\geq
1.$ (49)

Indeed, we have, using (12) and the rest of the properties of $\odot$ (all of
which can be traced back to Lemma 1.2.4 or the universal property of $\odot$,
[34, §2.1]),

$\displaystyle U_{k}\left({A_{0}\odot{\mathrm{Id}}_{n}^{\odot k-1}}\right)$
$\displaystyle=$
$\displaystyle\left[{\left({\sum_{j=0}^{k-1}{\mathrm{Id}}_{n}^{\odot j}\odot
F_{1}^{\odot
k-1-j}}\right)\odot\mbox{\boldmath${e}$\unboldmath}_{i}^{T}}\right]\left({A_{0}\odot{\mathrm{Id}}_{n}^{\odot
k-1}}\right)$ $\displaystyle=$
$\displaystyle\frac{k-1}{k}\left({\sum_{j=0}^{k-1}{\mathrm{Id}}_{n}^{\odot
j}\odot F_{1}^{\odot
k-1-j}}\right)\left({A_{0}\odot{\mathrm{Id}}_{n}^{k-2}}\right)\odot\mbox{\boldmath${e}$\unboldmath}_{i}^{T}+\frac{X_{i}}{k}\left({\sum_{j=0}^{k-1}{\mathrm{Id}}_{n}^{\odot
j}\odot F_{1}^{\odot k-j}}\right)$ $\displaystyle=$
$\displaystyle\frac{k-1}{k}\sum_{j=0}^{k-2}\frac{k-1-j}{k-1}F_{1}^{\odot
j}\odot A_{0}\odot{\mathrm{Id}}_{n}^{\odot
k-2-j}\odot\mbox{\boldmath${e}$\unboldmath}_{i}^{T}+\frac{X_{i}}{k}\left({\sum_{j=0}^{k-1}{\mathrm{Id}}_{n}^{\odot
j}\odot F_{1}^{\odot k-j}}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{k}\sum_{j=0}^{k-2}\left({k-1-j}\right)F_{1}^{\odot
j}\odot A_{0}\odot{\mathrm{Id}}_{n}^{\odot
k-2-j}\odot\mbox{\boldmath${e}$\unboldmath}_{i}^{T}+\frac{X_{i}}{k}\left({\sum_{j=0}^{k-1}{\mathrm{Id}}_{n}^{\odot
j}\odot F_{1}^{\odot k-j}}\right)$ $\displaystyle=$ $\displaystyle
X_{i}{\mathrm{Id}}_{n}^{\odot k-1}.$

The rest follows from (49):

$\displaystyle F_{k}\left({A_{0}\odot{\mathrm{Id}}_{n}^{\odot k-1}}\right)$
$\displaystyle=$
$\displaystyle-\frac{1}{X_{i}}\left[{\sum_{j=0}^{k-2}\binom{k-1}{j}F_{j+1}\left({A_{k-1-j}\odot{\mathrm{Id}}_{n}^{\odot
j}}\right)}\right]U_{k}\left({A_{0}\odot{\mathrm{Id}}_{n}^{k-1}}\right)$
$\displaystyle=$
$\displaystyle-\frac{1}{X_{i}}\left[{\sum_{j=0}^{k-2}\binom{k-1}{j}F_{j+1}\left({A_{k-1-j}\odot{\mathrm{Id}}_{n}^{\odot
j}}\right)}\right]X_{i}{\mathrm{Id}}_{n}^{\odot k-1}$ $\displaystyle=$
$\displaystyle-\sum_{j=0}^{k-2}\binom{k-1}{j}F_{j+1}\left({A_{k-1-j}\odot{\mathrm{Id}}_{n}^{\odot
j}}\right)$

which proves (47) true. ∎

The following corresponds to single-row blocks (i.e. one first integral at a
time) and is a valid precursor to future studies on how to maintain the
filtered admissible structure even after performing changes of variables, but
this is not immediately necessary to our current aims:

######  Conjecture 2.3.4.

Let $g$ be a first integral of (1) and $g^{\left({k}\right)}_{i}$ its
$k^{\mathrm{th}}$ lexicographic derivative, for every $k\geq 1$. Let
$g_{k}:=g^{\left({k}\right)}\left({\phi\left({t_{0}}\right)}\right)$ be its
value at the original point and define the matrix (constant, just like
$F_{k}$):

$M_{k}=\left({\begin{array}[]{ccccc}g_{1}^{\odot k}&&&\\\
c_{1,\dots,1,2}g_{1}^{\odot k-1}\odot g_{2}&\ddots&&\\\
\vdots&\ddots&g_{1}^{\odot 2}\\\
g_{k}&\cdots&g_{2}&g_{1}\end{array}}\right)\in\mathrm{Mat}_{1,n}^{n,n}$

be the first $k\times D_{n,k}$ block of $M:=\exp_{\odot}\left({\cdots
g_{k}\mid g_{k-1}\mid\dots\mid g_{2}\mid g_{1}}\right)$. Then

$\left({\Upsilon_{k}^{-1}}\right)^{T}\Phi_{k}^{T}M_{k}^{T}=\left({\begin{array}[]{cccc}\frac{1}{k!}\left({g^{k}}\right)^{\left({k}\right)}&\cdots&\frac{1}{2}\left({g^{2}}\right)^{\left({k}\right)}&g^{\left({k}\right)}\\\
&\ddots&\vdots&\vdots\\\
&&\frac{1}{2}\left({g^{2}}\right)^{\left({2}\right)}&g^{\left({2}\right)}\\\
&&&g^{\left({1}\right)}\end{array}}\right)$

In other words: $\left({\Upsilon^{-1}}\right)^{T}\Phi^{T}\Gamma^{T}$ is an
infinite matrix whose columns are the Taylor terms of the respective powers of
$g$. $\square$

######  Example 2.3.5.

In both cases (the quadrature-free one posed by Conjecture 2.3.1 and the
proven, quadrature-derived method in Theorem 2.2.2) every new Taylor block of
a given formal first integral can be obtained via a filter matrix; the only
difference is whether the matrix has a simple explicit form or not. Let us
show the first stages of _any_ filter matrix (proven or not) does to the
fundamental matrix. Let us assume, for instance, $i=1$. Stage $k=1$ is not
even a conjecture in virtue of Lemma 2.2.1 and the first order filter matrix,
namely $F_{1}$ in (41), transforms the transpose of the fundamental matrix of
the dual system $\Upsilon_{1}^{-1}=Y_{1}^{-1}$ into a matrix of the form

$\left({\;\begin{array}[]{|c|}\hline\cr\mbox{\boldmath${0}$\unboldmath}\\\
\hline\cr\rule{28.45274pt}{0.0pt}G_{1}\rule{28.45274pt}{0.0pt}\\\
\hline\cr\end{array}\;}\right):=\left({\;\begin{array}[]{?c?}\hrule
height=0.8pt\cr
width=0.8pt\lx@intercol\hfil\mbox{\boldmath${0}$\unboldmath}\hfil\lx@intercol\vrule
width=\\\ \hrule height=0.8pt\cr
width=0.8pt\lx@intercol\hfil\rule{28.45274pt}{0.0pt}g_{1}^{\left({1}\right)}\rule{28.45274pt}{0.0pt}\hfil\lx@intercol\vrule
width=\\\ \hrule height=0.1pt\cr width=0.8pt\lx@intercol\hfil
g_{2}^{\left({1}\right)}\hfil\lx@intercol\vrule width=\\\ \hrule
height=0.1pt\cr width=0.8pt\lx@intercol\hfil\vdots\hfil\lx@intercol\vrule
width=\\\ \hrule height=0.1pt\cr width=0.8pt\lx@intercol\hfil
g_{n-1}^{\left({1}\right)}\hfil\lx@intercol\vrule width=\\\ \hrule
height=0.8pt\cr\end{array}\;}\right)=\left({\begin{array}[]{ccccc}0&0&\cdots&0&0\\\
g_{1,1}^{\left({1}\right)}&g_{1,2}^{\left({1}\right)}&\cdots&g_{1,n-1}^{\left({1}\right)}&g_{1,n}^{\left({1}\right)}\\\
g_{2,1}^{\left({1}\right)}&g_{2,2}^{\left({1}\right)}&\cdots&g_{2,n-1}^{\left({1}\right)}&g_{2,n}^{\left({1}\right)}\\\
\vdots&\vdots&&\vdots&\vdots\\\
g_{n-1,1}^{\left({1}\right)}&g_{n-1,2}^{\left({1}\right)}&\cdots&g_{n-1,n-1}^{\left({1}\right)}&g_{n-1,n}^{\left({1}\right)}\\\
\end{array}}\right);$ (50)

each of these row vectors
$g_{1}^{\left({1}\right)},\dots,g_{n-1}^{\left({1}\right)}$ is the potential
gradient of a formal first integral and all $n-1$ of them are linearly
independent. In other words, they are the outputs of Lemma 2.2.1. The next
step would be to integrate the second variational system (from here onwards
only quadratures are needed):

$Y_{2}\left({t}\right)=\Phi_{1}\int_{0}^{t}\Phi_{1}^{-1}\left({\tau}\right)A_{2}\left({\tau}\right)\Phi_{1}^{\odot
2}\left({\tau}\right)d\tau$

invert the fundamental matrix for $\left({\mathrm{LVE}^{2}}\right)$,

$\Upsilon_{2}=\left({\begin{array}[]{cc}Y_{1}^{\odot 2}&0\\\
Y_{2}&Y_{1}\end{array}}\right),\qquad\mbox{thus}\qquad\Psi_{2}:=\left({\Phi_{2}^{-1}}\right)^{T}$

and multiply it by the second-order filter matrix

$\Phi_{2}=\left({\begin{array}[]{cc}F_{1}^{\odot 2}&0\\\
F_{2}&F_{1}\end{array}}\right),$

where $F_{2}$ is obtained from (45), (46) or from (43), and the result we
obtain is

$\Phi_{2}\left({\Psi_{2}}\right)^{T}=\Phi_{2}\Upsilon_{2}^{-1}=\left({\begin{array}[]{cc}G_{1}^{\odot
2}&0\\\
G_{2}&G_{1}\end{array}}\right)=\left({\begin{array}[]{cc}{\tiny\begin{array}[]{|c|}\hrule
height=0.5pt\cr\mbox{\boldmath${0}$\unboldmath}_{n}\\\ \hrule
height=0.1pt\cr\rule{12.80365pt}{0.0pt}g_{1}^{\left({1}\right)}\rule{12.80365pt}{0.0pt}\\\
\hrule height=0.1pt\cr\vdots\\\ \hrule height=0.1pt\cr
g_{n-1}^{\left({1}\right)}\\\ \hrule height=0.5pt\cr\end{array}\;}^{\odot
2}&\\\ \hrule
height=1.1pt\cr\vrule\lx@intercol\hfil\mbox{\boldmath${0}$\unboldmath}_{d_{n,2}}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\mbox{\boldmath${0}$\unboldmath}_{n}\hfil\lx@intercol\vrule\lx@intercol\\\
\hrule
height=0.1pt\cr\vrule\lx@intercol\hfil\rule{56.9055pt}{0.0pt}g_{1}^{\left({2}\right)}\rule{56.9055pt}{0.0pt}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\rule{22.76228pt}{0.0pt}g_{1}^{\left({1}\right)}\rule{22.76228pt}{0.0pt}\hfil\lx@intercol\vrule\lx@intercol\\\
\hrule
height=0.1pt\cr\vrule\lx@intercol\hfil\vdots\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\vdots\hfil\lx@intercol\vrule\lx@intercol\\\
\hrule height=0.1pt\cr\vrule\lx@intercol\hfil
g_{n-1}^{\left({2}\right)}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil
g_{n-1}^{\left({1}\right)}\hfil\lx@intercol\vrule\lx@intercol\\\ \hrule
height=1.1pt\cr\end{array}}\right)$ (51)

Take for instance the first non-zero row of the lower block $\left({G_{2}\mid
G_{1}}\right)$, i.e. the one highlighted in gray. Imagine we perform the
computations for $\left({\mathrm{LVE}_{\phi}^{3}}\right)$ and multiply
$\Upsilon_{3}^{-1}$ by the filter matrix $\Phi_{3}$; we obtain a matrix having
the one in (51) as a lower right $D_{n,k}\times D_{n,k}$ block:

$\displaystyle\Phi_{3}\Upsilon_{3}^{-1}\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\left({\begin{array}[]{ccc}F_{1}^{\odot 3}&0&0\\\ 3F_{1}\odot
F_{2}&F_{1}^{\odot}&0\\\
F_{3}&F_{2}&F_{1}\end{array}}\right)\Upsilon_{3}^{-1}=\left({\begin{array}[]{ccc}G_{1}^{\odot
3}&0&0\\\ 3G_{1}\odot G_{2}&G_{1}^{\odot}&0\\\
G_{3}&G_{2}&G_{1}\end{array}}\right)$ (58) $\displaystyle=$
$\displaystyle\left({\begin{array}[]{ccc}{\tiny\begin{array}[]{|c|}\hrule
height=0.5pt\cr\mbox{\boldmath${0}$\unboldmath}_{n}\\\ \hrule
height=0.1pt\cr\rule{12.80365pt}{0.0pt}g_{1}^{\left({1}\right)}\rule{12.80365pt}{0.0pt}\\\
\hrule height=0.1pt\cr\vdots\\\ \hrule height=0.1pt\cr
g_{n-1}^{\left({1}\right)}\\\ \hrule height=0.5pt\cr\end{array}\;}^{\odot
3}&&\\\ 3\;{\tiny\begin{array}[]{|c|}\hrule
height=0.5pt\cr\mbox{\boldmath${0}$\unboldmath}_{n}\\\ \hrule
height=0.1pt\cr\rule{12.80365pt}{0.0pt}g_{1}^{\left({1}\right)}\rule{12.80365pt}{0.0pt}\\\
\hrule height=0.1pt\cr\vdots\\\ \hrule height=0.1pt\cr
g_{n-1}^{\left({1}\right)}\\\ \hrule
height=0.5pt\cr\end{array}\;}\odot{\tiny\;\begin{array}[]{|c|}\hrule
height=0.5pt\cr\mbox{\boldmath${0}$\unboldmath}_{d_{n,2}}\\\ \hrule
height=0.1pt\cr\rule{28.45274pt}{0.0pt}g_{1}^{\left({2}\right)}\rule{28.45274pt}{0.0pt}\\\
\hrule height=0.1pt\cr\vdots\\\ \hrule height=0.1pt\cr
g_{n-1}^{\left({2}\right)}\\\ \hrule
height=0.5pt\cr\end{array}}&{\tiny\begin{array}[]{|c|}\hrule
height=0.5pt\cr\mbox{\boldmath${0}$\unboldmath}_{n}\\\ \hrule
height=0.1pt\cr\rule{12.80365pt}{0.0pt}g_{1}^{\left({1}\right)}\rule{12.80365pt}{0.0pt}\\\
\hrule height=0.1pt\cr\vdots\\\ \hrule height=0.1pt\cr
g_{n-1}^{\left({1}\right)}\\\ \hrule height=0.5pt\cr\end{array}\;}^{\odot
2}&\\\ \hrule
height=1.1pt\cr\vrule\lx@intercol\hfil\mbox{\boldmath${0}$\unboldmath}_{d_{n,3}}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\mbox{\boldmath${0}$\unboldmath}_{d_{n,2}}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\mbox{\boldmath${0}$\unboldmath}_{n}\hfil\lx@intercol\vrule\lx@intercol\\\
\hrule
height=0.1pt\cr\vrule\lx@intercol\hfil\rule{56.9055pt}{0.0pt}g_{1}^{\left({3}\right)}\rule{56.9055pt}{0.0pt}\hfil\lx@intercol&\vrule\lx@intercol\hfil\rule{56.9055pt}{0.0pt}g_{1}^{\left({2}\right)}\rule{56.9055pt}{0.0pt}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\rule{22.76228pt}{0.0pt}g_{1}^{\left({1}\right)}\rule{22.76228pt}{0.0pt}\hfil\lx@intercol\vrule\lx@intercol\\\
\hrule
height=0.1pt\cr\vrule\lx@intercol\hfil\vdots\hfil\lx@intercol&\vrule\lx@intercol\hfil\vdots\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\vdots\hfil\lx@intercol\vrule\lx@intercol\\\
\hrule height=0.1pt\cr\vrule\lx@intercol\hfil
g_{n-1}^{\left({n-1}\right)}\hfil\lx@intercol&\vrule\lx@intercol\hfil
g_{n-1}^{\left({2}\right)}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil
g_{n-1}^{\left({1}\right)}\hfil\lx@intercol\vrule\lx@intercol\\\ \hrule
height=1.1pt\cr\end{array}}\right)$ (81)

thus we have an increasing row, highlighted in darker gray color $\dots\mid
g_{1}^{\left({3}\right)}\mid g_{1}^{\left({2}\right)}\mid
g_{1}^{\left({1}\right)}$. These vectors are precisely the terms of a formal
Taylor series of a first integral
$g_{1}\left({\mbox{\boldmath${z}$\unboldmath}}\right)$ of (1) along solution
$\phi$:

$g_{1}\left({\mbox{\boldmath${z}$\unboldmath}+\phi}\right)=g_{1}\left({\phi}\right)+g_{1}^{\left({1}\right)}\mbox{\boldmath${z}$\unboldmath}+\frac{1}{2}g_{1}^{\left({2}\right)}\left({\phi}\right)\mbox{\boldmath${z}$\unboldmath}^{\odot
2}+\frac{1}{6}g_{1}^{\left({3}\right)}\left({\phi}\right)\mbox{\boldmath${z}$\unboldmath}^{\odot
3}+\dots$ (82)

where
$g_{1}^{\left({i}\right)}=g_{1}^{\left({i}\right)}\left({\mbox{\boldmath${\phi}$\unboldmath}}\right)$,
i.e. the implementation of (3) in 1.2.1 (2.) to
$\mbox{\boldmath${z}$\unboldmath}=\phi$. The same applies, _mutatis mutandis_
, to the rest of rows of infinite block $\dots G_{3}\mid G_{2}\mid G_{1}$,
i.e. replacing $g^{\left({i}\right)}_{1}$ with $g^{\left({i}\right)}_{j}$ for
some other $j=2,\dots,n-1$. Needless to say the upper terms $...$ in (58) and
higher-order counterparts contain all the cross-products arising from
derivatives of products of the first integrals, e.g.
$g_{1}^{\left({1}\right)}\odot g_{3}^{\left({2}\right)}$ would be a row in
$G_{1}\odot G_{2}$ (it is easy to check which one by way of lexicographic
ordering and we invite the reader to do this themselves, using the examples in
§3).

##  3 Examples

###  3.1 Dixon’s system

The following system,

$\dot{u}=\frac{uv}{u^{2}+v^{2}}-\alpha
u,\qquad\dot{v}=\frac{v^{2}}{u^{2}+v^{2}}-\beta v+\beta-1,$ (83)

where $\alpha,\beta>0$, arose in [8] as a decoupled two-dimensional fragment
of a three-dimensional dynamical model of the magnetic field of neutron stars.
Several references (e.g. [2, 11, 37]) discussed the ostensible chaotic
behavior of the dynamics of (83). [33], on the other hand, purported the non-
chaotic character of this system based on a version of the Poincaré-Bendixson
Theorem that precludes the need for compact sets comprising finitely many sets
of homoclinic/heteroclinic connections. Given the existing, seldom-discussed
gap between the concepts _non-chaotic_ and _integrable_ (regardless of which
ad-hoc definition of chaos is used), we can afford to eschew the polemic
altogether and focus on finding a conserved quantity for (83) using higher
variational equations.

We will focus on the case $\alpha=\beta>0$. The case $\alpha\neq\beta$, as
well as the original three-dimensional model ([11, eqs (3)-(5)]), will be
tackled in future work.

$\alpha=1$ is immediate to solve and needs no further consideration. Let us
focus on $\alpha\neq 1$. Previous attempts at simplifying higher-order
$\mathrm{LVE}^{k}$ suggest change of variable

$\left({u,v}\right)=\left({\frac{\sin y}{x},\frac{\cos y}{x}}\right)$ (84)

which transforms (83) for $\alpha=\beta$ into

$\dot{x}=\alpha x\left({1-x\cos
y}\right),\qquad\dot{y}=-\left({\alpha-1}\right)x\sin y.$ (85)

We will use particular solution

$\phi\left({t}\right)=\left({x\left({t}\right),0}\right)\qquad\mbox{ such that
}\dot{x}=-\alpha\left({-1+x}\right)x.$ (86)

Along $\Gamma$, the higher variational complex and its dual
($\mathrm{LVE}_{\phi}^{\star}$) feature the following blocks for $k\geq 0$,

$A_{k}=\left\\{\begin{array}[]{ll}X\left({x,y}\right)=\left({\begin{array}[]{c}\alpha
x\left({1-x\cos y}\right)\\\ -\left({\alpha-1}\right)x\sin
y\end{array}}\right),&k=0,\\\
\left(\begin{array}[]{cc}-\alpha\left({2x-1}\right)&0\\\
0&-\left({\alpha-1}\right)x\\\ \end{array}\right),&k=1,\\\
\left({\begin{array}[]{cccccc}\cline{1-3}\cr\vrule\lx@intercol\hfil\hbox{\multirowsetup$0_{2\times\left({k-2}\right)}$}\hfil\lx@intercol\vrule\lx@intercol&\left({-1}\right)^{k/2}2\alpha&0&\left({-1}\right)^{k/2+1}\alpha
x^{2}\\\
\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol&0&\left({-1}\right)^{k/2}\left({-1+\alpha}\right)&0\\\
\cline{1-3}\cr\end{array}}\right),&k\mbox{ even}\\\ &\\\
\left({\begin{array}[]{ccccc}\cline{1-3}\cr\vrule\lx@intercol\hfil\hbox{\multirowsetup$0_{2\times\left({k-1}\right)}$}\hfil\lx@intercol\vrule\lx@intercol&\left({-1}\right)^{\frac{k+1}{2}}2\alpha
x&0\\\
\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol&0&\left({-1}\right)^{\frac{k+1}{2}}\left({-1+\alpha}\right)x\\\
\cline{1-3}\cr\end{array}}\right),&k\mbox{ odd},\end{array}\right.$ (87)

for instance a principal fundamental matrix of ($\mathrm{VE}_{\phi}$) is

$\Upsilon_{1}=Y_{1}=\left(\begin{array}[]{cc}\frac{(x-1)x}{(x_{0}-1)x_{0}}&0\\\
0&(1-x_{0})^{\frac{1}{\alpha}-1}(1-x)^{1-\frac{1}{\alpha}}\\\
\end{array}\right)$

and the first-order filter matrix is very simple in this case,

$F_{1}={\mathrm{Id}}_{2}-\frac{1}{X_{1}^{0}}\left({X^{0}\odot\mbox{\boldmath${e}$\unboldmath}_{1}^{T}}\right)=\left(\begin{array}[]{cc}0&0\\\
-\frac{X_{2}\left({x_{0},y_{0}}\right)}{X_{1}\left({x_{0},y_{0}}\right)}&1\\\
\end{array}\right)=\left(\begin{array}[]{cc}0&0\\\ 0&1\\\ \end{array}\right),$

where $x_{0}=x\left({0}\right)$ and $y_{0}=y\left({0}\right)$; so are the
higher-order matrices in $\Phi=\exp_{\odot}F$ obtained from (45), some of
which are shown below:

$\displaystyle F_{2}=\left(\begin{array}[]{ccc}0&0&0\\\
0&\frac{1-\alpha}{\alpha(x_{0}-1)}&0\\\ \end{array}\right),\quad
F_{3}=\left(\begin{array}[]{cccc}0&0&0&0\\\
0&\frac{(\alpha-1)(2\alpha-1)}{\alpha^{2}(x_{0}-1)^{2}}&0&0\\\
\end{array}\right),$ $\displaystyle
F_{4}=\left(\begin{array}[]{ccccc}0&0&0&0&0\\\
0&-\frac{(\alpha-1)(2\alpha-1)(3\alpha-1)}{\alpha^{3}(x_{0}-1)^{3}}&0&-\frac{(\alpha-1)(2x_{0}+1)}{\alpha(x_{0}-1)^{2}}&0\\\
\end{array}\right),\dots$

Returning to $k=1$, Lemma 2.2.1 ensures that the second column of the filtered
matrix

$\left({Y_{1}^{-1}}\right)^{T}F_{1}^{T}=\left(\begin{array}[]{cc}0&0\\\
0&(1-x_{0})^{1-\frac{1}{\alpha}}(1-x)^{\frac{1}{\alpha}-1}\\\
\end{array}\right),$

is an admissible solution of degree $1$. We can normalize this to eliminate
appearances of $x_{0}$:

$\left({\Phi_{1}^{-1}}\right)^{T}F_{1}^{T}P^{T}:=\left(\begin{array}[]{cc}0&0\\\
0&(1-x_{0})^{1-\frac{1}{\alpha}}(1-x)^{\frac{1}{\alpha}-1}\\\
\end{array}\right)\left(\begin{array}[]{cc}1&0\\\
0&(1-x_{0})^{\frac{1}{\alpha}-1}\\\
\end{array}\right)^{T}=\left(\begin{array}[]{c|c|}\cline{2-2}\cr 0&0\\\
0&(1-x)^{\frac{1}{\alpha}-1}\\\ \cline{2-2}\cr\end{array}\right),$

the second of whose columns is still the gradient $f^{\left({1}\right)}$ of a
formal first integral (this is exactly what appears as a row
$g_{1}^{\left({1}\right)}$ in (50)).

The higher order trimming (in which we discard redundant symmetric products
$f^{\left({i_{1}}\right)}\odot\cdots\odot f^{\left({i_{k}}\right)}$, i.e.
derivatives of powers of $f$) is clear once we write down the dual system of
equations and filter out telescopically using either Lemma 2.1.1 and the
general form of that each new term $f^{\left({k}\right)}$ of our formal first
integral $f$ is expressible as the following (bear in mind that
$x=x\left({t}\right)$ here):
$f^{\left({k}\right)}\left({x}\right)=\left({\frac{d}{dx}f^{\left({k-1}\right)}\mid
g_{k}\left({x}\right)}\right)$, where

$\sum_{j=0}^{\frac{k-1}{2}}\left({\left({-1}\right)^{j+1}\left({\alpha-1}\right)\binom{k}{-1-2j+k}g_{k-2j}\left({x}\right)+\left({-1}\right)^{j+1}\alpha\binom{k}{-2j+k}xg_{k-2j}^{\prime}\left({x}\right)}\right)+\alpha
g_{k}^{\prime}\left({x}\right)=0,$

which means $g_{k}\left({x}\right)=0$ for every even $k\geq 2$ and e.g.

$\displaystyle f_{1}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}0&(1-x)^{\frac{1}{\alpha}-1}\\\
\end{array}\right),$ $\displaystyle f_{2}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}0&\frac{(\alpha-1)(1-x)^{\frac{1}{\alpha}-2}}{\alpha}&0\\\
\end{array}\right),$ $\displaystyle f_{3}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}0&\left(\frac{1}{\alpha}-2\right)\left(\frac{1}{\alpha}-1\right)(1-x)^{\frac{1}{\alpha}-3}&0&\frac{(1-x)^{\frac{1}{\alpha}-2}(2\alpha+(\alpha-2)x-1)}{\alpha-2}\\\
\end{array}\right),$ $\displaystyle f_{4}$ $\displaystyle=$
$\displaystyle\left({\begin{array}[]{c|c}\frac{d}{dx}f_{3}&0\end{array}}\right),$
$\displaystyle f_{5}$ $\displaystyle=$
$\displaystyle\left({\begin{array}[]{c|c}\frac{d}{dx}f_{4}&-\frac{(1-x)^{\frac{1}{\alpha}-3}(-(\alpha-2)x((\alpha-2)(3\alpha-4)x+\alpha(39\alpha-55)+14)+\alpha(2(53-24\alpha)\alpha-73)+16)}{(\alpha-2)^{2}(3\alpha-4)}\end{array}}\right),$

This can be achieved via (43) in Theorem 2.2.2 (the easiest option even for
low values of $k$) or the filter matrix described in §2.3 to compute each
$f_{k}$ in terms of the preceding $f_{k-1},\dots,f_{1}$. We invite the reader
to apply either method and realize that the above terms appear with only minor
attention paid at each step.

A tedious scrutiny of the general form of these vectors shows that they are
precisely limit cases of the higher-order Taylor terms (along
$\left({x\left({t}\right),0}\right)\in\Gamma$) of the following function:

$f=\left(x\sin^{\frac{\alpha}{1-\alpha}}(y)-\left({\frac{\alpha\cos
y}{\alpha-1}}\right){}_{2}F_{1}\left(\frac{1}{2},\frac{\alpha}{2(\alpha-1)}+1;\frac{3}{2};\cos^{2}y\right)\right){}^{\frac{1}{\alpha}-1}$

where ${}_{2}F_{1}$ is the Gaussian hypergeometric function [1]. Removing the
outer power yields a first integral of (85) as well, as can be easily checked:

$f=x\sin^{\frac{\alpha}{1-\alpha}}(y)-\left({\frac{\alpha\cos
y}{\alpha-1}}\right){}_{2}F_{1}\left(\frac{1}{2},\frac{\alpha}{2(\alpha-1)}+1;\frac{3}{2};\cos^{2}y\right).$

Undoing the transformation (84) we obtain:

######  Theorem 3.1.1.

The following function

$F=u^{\frac{\alpha}{1-\alpha}}v^{\frac{1}{\alpha-1}}\left(\frac{u^{2}}{v^{2}}+1\right)^{\frac{1}{2(\alpha-1)}}-\frac{\alpha
v\,_{2}F_{1}\left(\frac{1}{2},\frac{\alpha}{2(\alpha-1)}+1;\frac{3}{2};\frac{v^{2}}{u^{2}+v^{2}}\right)}{(\alpha-1)\sqrt{u^{2}+v^{2}}}$
---

is a first integral of the original special CDK system. $\square$

###  3.2 The SIR model with vital dynamics

The SIR epidemiological model with vital dynamics [10, 17] is given by

$\left\\{\begin{array}[]{lcl}\dot{S}&=&\mu(n-S)-\frac{\beta SI}{n},\\\
\dot{I}&=&\frac{\beta SI}{n}-I(\gamma+\mu),\\\ \dot{R}&=&\gamma I-\mu
R.\end{array}\right.$ (95)

The system was first introduced in [17] strictly for the modelling of
infectious diseases, but has since been applied to a number of situations
allowing dynamic compartmentalization, e.g. marketing [32]. The study of its
integrability has been the subject of several works already ([6, 14]) and the
search for a first integral is still an open problem. We will address specific
examples; the general case will be addressed in the future.

First, let us start with a very simple example whose solution we already know:
$\beta=\gamma$ and $\mu=0$ and change of variable

$S=n\left({1+xy}\right),\quad I=-ny,\quad R=\gamma nz,$ (96)

we obtain system of equations
$\frac{d}{dt}\left({x,y,z}\right)=\left({\gamma-\gamma(x-1)xy,\gamma
xy^{2},-y}\right)$, and applying Conjectures 2.3.1, 2.3.4 and (if the matrix
is used) Lemma 2.1.1, the general terms of the jets telegraph a very
identifiable structure: that of two first integrals for the transformed
system, $g^{\star}=\frac{\log(xy+1)}{\gamma}+z$, $f^{\star}=-xy+\log(xy+1)+y$
which transform back by (96) into
$g=\frac{n\log\left(\frac{S}{n}\right)+R}{\gamma n}$,
$f=-\frac{S+I}{n}+\log\left(\frac{S}{n}\right)+1$, thus taking us to the
already-known first-integral $S+I+R-n$ (thus, $S+I+R$) and the remaining one,
$\frac{n\log\left(\frac{S}{n}\right)+R}{\gamma n}$ (alternatively,
$\frac{S}{n}e^{\frac{R}{n}}$). This is all in keeping with what is already
known about the integrability of the system for $\mu=0$ (see e.g. [6, §1]).
The reader can fill in the blanks for filter matrices or progressive filtering
(Theorem 2.2.2) and formal Taylor terms here; we will add more detail about
these in the less trivial case below.

To wit, the case in which $\gamma=0$ and $\beta,\mu\neq 0$. To the best of our
knowledge, the integrability of this case is an open problem in earlier
references. Change of variables

$\left({S,I,R}\right)=\left({y^{2}\left(x^{-\frac{\beta}{\mu}}+1\right)+n,-x^{-\frac{\beta}{\mu}}y^{2},y^{2}z}\right),$
(97)

itself suggested by an earlier try at simplifying $\mathrm{LVE}^{3}_{\phi}$,
provides us with new system

$\dot{x}=-\frac{\mu
x\left[y^{2}\left(x^{-\frac{\beta}{\mu}}+1\right)+n\right]}{n},\quad\dot{y}=-\frac{1}{2}\mu
y,\quad\dot{z}=0.$ (98)

This system admits an invariant plane $y=z=0$ containing particular solution

$\phi\left({t}\right)=\left({\widehat{x}\left({t}\right),0,0}\right),\qquad\mbox{where
}\frac{d}{dt}\widehat{x}=-\mu\widehat{x}.$ (99)

This leads to the variational system
$\dot{Y}=\left({A\odot\exp_{\odot}{\mathrm{Id}}_{n}}\right)Y$ along $\phi$
where

$A_{\mathrm{LVE}_{\phi}}=A\odot\exp_{\odot}{\mathrm{Id}}_{n}=\left({\begin{array}[]{ccccc}\ddots&&&&\\\
\cdots&4A_{1}\odot{\mathrm{Id}}_{n}^{\odot 3}&&&\\\
\cdots&6A_{2}\odot{\mathrm{Id}}_{n}^{\odot
2}&3A_{1}\odot{\mathrm{Id}}_{n}^{\odot 2}&&\\\
\cdots&4A_{3}\odot{\mathrm{Id}}_{n}&3A_{2}\odot{\mathrm{Id}}_{n}&2A_{1}\odot{\mathrm{Id}}_{n}&\\\
\cdots&A_{4}&A_{3}&A_{2}&A_{1}\end{array}}\right),$

and $A_{i}$ defined per (27) appear as follows:

$\displaystyle A_{0}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}-\mu\widehat{x}(t)\\\ 0\\\ 0\\\
\end{array}\right)$ $\displaystyle A_{1}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}-\mu&0&0\\\ 0&-\frac{\mu}{2}&0\\\
0&0&0\\\ \end{array}\right)$ $\displaystyle A_{2}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccccc}0&0&0&\frac{2\mu\widehat{x}(t)\left(-\widehat{x}(t)^{-\frac{\beta}{\mu}}-1\right)}{n}&0&0\\\
0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\ \end{array}\right)$ $\displaystyle A_{3}$
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccccccccc}0&0&0&\frac{2\left((\beta-\mu)\widehat{x}(t)^{-\frac{\beta}{\mu}}-\mu\right)}{n}&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0\\\ \end{array}\right)$
$\displaystyle A_{k}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccccccc}\cline{5-7}\cr
0&0&0&\frac{2\left((\beta-\mu)\widehat{x}(t)^{-\frac{\beta}{\mu}}-\mu\right)}{n}&\vrule\lx@intercol\hfil\hbox{\multirowsetup$0_{3\times
d_{3,k-4}}$}\hfil\lx@intercol\vrule\lx@intercol\\\
0&0&0&0&\vrule\lx@intercol\hfil\rule{28.45274pt}{0.0pt}\hfil\lx@intercol\vrule\lx@intercol\\\
0&0&0&0&\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol\\\
\cline{5-7}\cr\end{array}\right),\qquad k\geq 4$

Lower blocks of the quadrature-free filter matrix are built as proposed in
§2.3, let us write the first few cases

$F_{1}\\!=\\!\left(\begin{array}[]{ccc}0&0&0\\\ 0&1&0\\\ 0&0&1\\\
\end{array}\right),\,F_{2}\\!=\\!\left(\\!\begin{array}[]{ccc}\cline{3-3}\cr
0&0&\vrule\lx@intercol\hfil\hbox{\multirowsetup\\!$0_{3\times
4}$\\!}\hfil\lx@intercol\vrule\lx@intercol\\\
0&-\frac{1}{2\widehat{x}_{0}}&\vrule\lx@intercol\hfil\rule{0.0pt}{0.0pt}\hfil\lx@intercol\vrule\lx@intercol\\\
0&0&\vrule\lx@intercol\hfil\rule{0.0pt}{0.0pt}\hfil\lx@intercol\vrule\lx@intercol\\\
\cline{3-3}\cr\end{array}\right),\,F_{3}\\!=\\!\left(\begin{array}[]{ccc}\cline{3-3}\cr
0&0&\vrule\lx@intercol\hfil\hbox{\multirowsetup\\!$0_{3\times
8}$\\!}\hfil\lx@intercol\vrule\lx@intercol\\\
0&\frac{3}{4\widehat{x}_{0}^{2}}&\vrule\lx@intercol\hfil\rule{14.22636pt}{0.0pt}\hfil\lx@intercol\vrule\lx@intercol\\\
0&0&\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol\\\
\cline{3-3}\cr\end{array}\right),\,F_{4}\\!=\\!\left(\begin{array}[]{ccccc}\cline{3-3}\cr\cline{5-5}\cr
0&0&\vrule\lx@intercol\hfil\hbox{\multirowsetup\\!$0_{3\times
4}$\\!}\hfil\lx@intercol\vrule\lx@intercol&0&\vrule\lx@intercol\hfil\hbox{\multirowsetup\\!$0_{3\times
8}$\\!}\hfil\lx@intercol\vrule\lx@intercol\\\
0&-\frac{15}{8\widehat{x}_{0}^{3}}&\vrule\lx@intercol\hfil\rule{14.22636pt}{0.0pt}\hfil\lx@intercol\vrule\lx@intercol&\frac{3\left(\widehat{x}_{0}^{-\frac{\beta}{\mu}}+1\right)}{n\widehat{x}_{0}}&\vrule\lx@intercol\hfil\rule{14.22636pt}{0.0pt}\hfil\lx@intercol\vrule\lx@intercol\\\
0&0&\vrule\lx@intercol\hfil\rule{14.22636pt}{0.0pt}\hfil\lx@intercol\vrule\lx@intercol&0&\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol\\\
\cline{3-3}\cr\cline{5-5}\cr\end{array}\right).$

The filtered matrix, has two linearly independent admissible solutions as its
nonzero rows:

$F_{1}Y_{1}^{-1}=\left(\begin{array}[]{ccc}0&0&0\\\
0&\frac{\sqrt{\widehat{x}_{0}}}{\sqrt{\widehat{x}(t)}}&0\\\ 0&0&1\\\
\end{array}\right),$

which (optionally) multiplied by

$P_{1}=\left(\begin{array}[]{ccc}1&0&0\\\
0&\frac{1}{\sqrt{\widehat{x}_{0}}}&0\\\ 0&0&1\\\ \end{array}\right)$ (105)

removes occurrences of the initial condition $\widehat{x}_{0}$:

$G_{1}=\left(\begin{array}[]{ccc}0&0&0\\\
\cline{1-3}\cr\vrule\lx@intercol\hfil
0\hfil\lx@intercol&\frac{1}{\sqrt{\widehat{x}(t)}}&\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol\\\ \cline{1-3}\cr\vrule\lx@intercol\hfil
0\hfil\lx@intercol&0&\lx@intercol\hfil 1\hfil\lx@intercol\vrule\lx@intercol\\\
\cline{1-3}\cr\end{array}\right)=\left(\begin{array}[]{ccc}0&0&0\\\
\cline{1-3}\cr\vrule\lx@intercol\hfil
f^{\left({1}\right)}\hfil\lx@intercol\vrule\lx@intercol\\\
\cline{1-3}\cr\vrule\lx@intercol\hfil
g^{\left({1}\right)}\hfil\lx@intercol\vrule\lx@intercol\\\
\cline{1-3}\cr\end{array}\right).$

Let us write things down for $k=3$: once we have solved the direct variational
system $\left({\mathrm{LVE}_{\Gamma}^{3}}\right)$ and obtained its fundamental
matrix $\Upsilon_{3}$, the principal fundamental matrix of the dual system is
nothing but the transpose of its inverse; we can either keep said transpose
$\left({\Upsilon_{3}^{-1}}\right)^{T}$ and right-multiply it by proposed
filter $\Phi_{3}^{T}$ (in which case first integral jets appear as columns) or
left-multiply the original $\Upsilon_{3}^{-1}$ by $\Phi_{3}$ in which case
jets appear as rows in the bottom block:

$\displaystyle\Phi_{3}\Upsilon_{3}^{-1}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}F_{1}^{\odot 3}&0&0\\\ 3F_{1}\odot
F_{2}&F_{1}^{\odot 2}&0\\\ F_{3}&F_{2}&F_{1}\\\
\end{array}\right)\left(\begin{array}[]{ccc}Y_{1}^{\odot 3}&0&0\\\ 3Y_{1}\odot
Y_{2}&Y_{1}^{\odot 2}&0\\\ Y_{3}&Y_{2}&Y_{1}\\\ \end{array}\right)^{-1}$
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccccccccccccccccccc}\hline\cr\vrule\lx@intercol\hfil\hbox{\multirowsetup$0_{6\times
19}$}\hfil\lx@intercol\vrule\lx@intercol\\\
\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol\\\ \hline\cr
0&0&0&0&0&0&\frac{\widehat{x}_{0}^{3/2}}{\widehat{x}^{3/2}}&0&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&\frac{\widehat{x}_{0}}{\widehat{x}}&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&\frac{\sqrt{\widehat{x}_{0}}}{\sqrt{\widehat{x}}}&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&1&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&-\frac{\widehat{x}_{0}}{\widehat{x}^{2}}&0&0&0&0&0&0&0&0&0&\frac{\widehat{x}_{0}}{\widehat{x}}&0&0&0&0&0\\\
0&0&0&0&-\frac{\sqrt{\widehat{x}_{0}}}{2\widehat{x}^{3/2}}&0&0&0&0&0&0&0&0&0&\frac{\sqrt{\widehat{x}_{0}}}{\sqrt{\widehat{x}}}&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
0&\frac{3\sqrt{\widehat{x}_{0}}}{4\widehat{x}^{5/2}}&0&0&0&0&a_{18,7}&0&0&0&0&-\frac{\sqrt{\widehat{x}_{0}}}{2\widehat{x}^{3/2}}&0&0&0&0&0&\frac{\sqrt{\widehat{x}_{0}}}{\sqrt{\widehat{x}}}&0\\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1\\\ \end{array}\right)$

where $\widehat{x}_{0}=\widehat{x}\left({0}\right)$ and

$a_{18,7}=\frac{\widehat{x}(t)^{-\frac{\beta}{\mu}-\frac{3}{2}}\left(3\mu\widehat{x}(t)^{\beta/\mu}\widehat{x}_{0}^{\frac{3}{2}-\frac{\beta}{\mu}}+3(\mu-\beta)\sqrt{\widehat{x}_{0}}(\widehat{x}_{0}-\widehat{x}(t))\widehat{x}(t)^{\beta/\mu}-3\mu\sqrt{\widehat{x}_{0}}\widehat{x}(t)\right)}{(\beta-\mu)n}$

The bottom two rows are already admissible jets
$\star^{\left({3}\right)}\mid\star^{\left({2}\right)}\mid\star^{\left({1}\right)}$,
but we can trim them further by multiplying this matrix by

$\Pi_{3}=\left(\begin{array}[]{ccc}P_{1}^{\odot 3}&0&0\\\ 3P_{1}\odot
P_{2}&P_{1}^{\odot 2}&0\\\ P_{3}&P_{2}&P_{1}\\\ \end{array}\right)^{T}$

where $P_{1}$ is as in (105) and $P_{2},P_{3}$ are successively chosen so as
vanish $\widehat{x}_{0}$ from the matrix (in this case, all terms in
$P_{2},P_{3}$ equal to zero except for
$\left({P_{3}}\right)_{2,7}=\frac{3\left(\mu\left(-\widehat{x}_{0}^{-\frac{\beta}{\mu}}-1\right)+\beta\right)}{n\sqrt{\widehat{x}_{0}}(\beta-\mu)}$).
This equates to weeding out copies of successive symmetric products of
$f^{\left({i}\right)}$, and/or $g^{\left({j}\right)}$ and harks back to Lemma
2.1.1 and Conjecture 2.3.4. The resulting matrix retains its neat echeloned
structure

$\Pi_{3}\Phi_{3}\Upsilon_{3}^{-1}=\left(\begin{array}[]{ccc}G_{1}^{\odot
3}&0&0\\\ 3G_{1}\odot G_{2}&G_{1}^{\odot 2}&0\\\ G_{3}&G_{2}&G_{1}\\\
\end{array}\right)$

where lower block $G_{3}\mid G_{2}\mid G_{1}$ has independent formal first
integral jets as its non-zero rows:

$\left(\begin{array}[]{ccccccccccccccccccc}0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
\hline\cr\vrule\lx@intercol\hfil
0\hfil\lx@intercol&\frac{3}{4\widehat{x}^{5/2}}&0&0&0&0&\frac{3\left(-\mu\widehat{x}^{-\frac{\beta}{\mu}}+\beta-\mu\right)}{(\beta-\mu)n\sqrt{\widehat{x}}}&0&0&0&\vrule\lx@intercol\hfil
0\hfil\lx@intercol&-\frac{1}{2\widehat{x}^{3/2}}&0&0&0&0&\vrule\lx@intercol\hfil
0\hfil\lx@intercol&\frac{1}{\sqrt{\widehat{x}}}&\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol\\\ \hline\cr\vrule\lx@intercol\hfil
0\hfil\lx@intercol&0&0&0&0&0&0&0&0&0&\vrule\lx@intercol\hfil
0\hfil\lx@intercol&0&0&0&0&0&\vrule\lx@intercol\hfil
0\hfil\lx@intercol&0&\lx@intercol\hfil 1\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr\end{array}\right)$ (108)

in other words, the following are degree-three asymptotic expressions of
formal first integrals
($\mbox{\boldmath${z}$\unboldmath}=\left({x,y,z}\right)$ and
$\phi=\left({\widehat{x},0,0}\right)$ is the particular solution as usual):

$\displaystyle f\left({\mbox{\boldmath${z}$\unboldmath}+\phi}\right)$
$\displaystyle=$ $\displaystyle
f^{\left({1}\right)}\left({\phi}\right)\mbox{\boldmath${z}$\unboldmath}+\frac{1}{2}f^{\left({2}\right)}\left({\phi}\right)\mbox{\boldmath${z}$\unboldmath}^{\odot
2}+\frac{1}{3!}f^{\left({3}\right)}\mbox{\boldmath${z}$\unboldmath}^{\odot
3}+O\left({\mbox{\boldmath${z}$\unboldmath}^{\odot 4}}\right)$
$\displaystyle=$
$\displaystyle\frac{\large{y}}{\sqrt{\widehat{x}}}-\frac{x\large{y}}{2\widehat{x}^{3/2}}+\frac{1}{6}\left(\frac{3\large{y}^{3}\left(\frac{\mu\widehat{x}^{1-\frac{\beta}{\mu}}}{\mu-\beta}+\widehat{x}\right)}{n\widehat{x}^{3/2}}+\frac{9x^{2}\large{y}}{4\widehat{x}^{5/2}}\right)+O\left({\mbox{\boldmath${z}$\unboldmath}^{\odot
4}}\right),$ $\displaystyle
g\left({\mbox{\boldmath${z}$\unboldmath}+\phi}\right)$ $\displaystyle=$
$\displaystyle
g^{\left({1}\right)}\left({\phi}\right)\mbox{\boldmath${z}$\unboldmath}+\frac{1}{2}g^{\left({2}\right)}\left({\phi}\right)\mbox{\boldmath${z}$\unboldmath}^{\odot
2}+\frac{1}{3!}g^{\left({3}\right)}\mbox{\boldmath${z}$\unboldmath}^{\odot
3}+O\left({\mbox{\boldmath${z}$\unboldmath}^{\odot
4}}\right)=z+O\left({\mbox{\boldmath${z}$\unboldmath}^{\odot 4}}\right).$

Again, Theorem 2.2.2 does the exact same job faster and with the guarantee
that it works, e.g.

$\displaystyle f^{\left({2}\right)}$ $\displaystyle=$
$\displaystyle-\left[{\left({Y_{1}^{-1}}\right)^{\odot
2}}\right]^{T}\left[{\int_{\widehat{x}_{0}}^{\widehat{x}}\left({Y_{1}^{\odot
2}}\right)^{T}f^{\left({1}\right)}A_{2}dx-\left({0,\frac{1}{2\widehat{x}_{0}^{3/2}},0,0,0,0}\right)}\right],$
$\displaystyle f^{\left({3}\right)}$ $\displaystyle=$
$\displaystyle-\left[{\left({Y_{1}^{-1}}\right)^{\odot
3}}\right]^{T}\left[{\int_{\widehat{x}_{0}}^{\widehat{x}}\left({Y_{1}^{\odot
3}}\right)^{T}\left[{3f^{\left({2}\right)}\left({A_{2}\odot{\mathrm{Id}}_{3}}\right)+f^{\left({1}\right)}A_{3}}\right]dx-\left({0,-\frac{3}{4\widehat{x}_{0}^{5/2}},0,0,0,0,0,0,0,0}\right)}\right],$

and the reader can check that the jet terms are exactly the same as those in
(108). Either method may sound like a cumbersome process but it only requires
special attention in its first stages if savvy simplifications are chosen, as
was the case for Dixon’s system. Later stages of the bottom row remain zero
and accordingly $g=z$ is easily checked to be a first integral (formal and
convergent) for (98); for $f$, and much like in §3.1, pattern checks along
with simple resummation yield a first integral for system (98),

$f=\frac{ye^{\frac{y^{2}}{2n}}}{\left[e^{\frac{uy^{2}}{n}}\left(\frac{uy^{2}}{n}\right)^{u}\Gamma\left(1-u,\frac{uy^{2}}{n}\right)+x^{u}\right]^{\frac{1}{2u}}},$

where $u=\frac{\beta}{\mu}$ and $\Gamma\left({\cdot,\cdot}\right)$ is the
incomplete Gamma function [1]; undoing transformation (97) we obtain something
that is now immediate to prove on its own:

######  Theorem 3.2.1.

The following two functions,

$\displaystyle F$ $\displaystyle=$
$\displaystyle\frac{e^{\frac{-n+S+I}{2n}}\sqrt{-n+S+I}}{\left(\frac{-n+S+I}{I}-\left(\frac{\beta}{\mu
n}\right)^{\beta/\mu}e^{\frac{\beta(-n+S+I)}{\mu
n}}(-n+S+I)^{\beta/\mu}\Gamma\left(1-\frac{\beta}{\mu},\frac{\beta(-n+S+I)}{\mu
n}\right)\right)^{\frac{\mu}{2\beta}}},$ $\displaystyle G$ $\displaystyle=$
$\displaystyle\frac{R}{-n+S+I}.$

are two functionally independent first integrals of SIR with $\gamma=0$ (95).
$\square$

It is worth noting, however, that the only easy goal we can aspire to in most
cases will be that of formal first integrals with a compact identifiable
general term but without a straightforward convergent expression, and
resummation and deep pattern recognition techniques will need to be adapted to
each problem. This will also be the subject of future research in the short
term but for the time being let us see a short example below.

###  3.3 The van der Pol oscillator

This well-known system is usually presented as an epitome of the proverbial
“simple yet chaotic” dynamical system [37]

$\ddot{u}=\mu(1-u^{2})\dot{u}-u$ (109)

Value $\mu=2$ appears to have a significant, yet still unclarified role in
potential qualitative simplifications in variable transformations. Thus let us
fix this value for the present paper:

$\dot{u}=v,\qquad\dot{v}=2(1-u^{2})v-u$ (110)

Perform the change of variables

$\left({u\left({t}\right),v\left({t}\right)}\right)=\left({\frac{x\left({t}\right)y\left({t}\right)}{\sqrt{2}},\left(\frac{x\left({t}\right)}{\sqrt{2}}-\frac{1}{\sqrt{2}}\right)y\left({t}\right)}\right)$
(111)

that will simplify the corresponding jet, and our transformed system is

$\dot{x}=-(x-1)x^{3}y^{2}-1,\qquad\dot{y}=(x-1)x^{2}y^{3}+y.$ (112)

The dual infinite variational system
$\left({\mathrm{VE}_{\Gamma}}\right)^{\star}$ along particular solution

$\Gamma=\left\\{{\left({x\left({t}\right),0}\right):t\in{\mathbb{C}}}\right\\},\qquad\mbox{where}\quad\dot{x}\left({t}\right)=-1,$
(113)

has the following very simple matrices:

$\displaystyle A_{1}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}0&0\\\ 0&1\\\ \end{array}\right),$
$\displaystyle A_{2}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}0&0&-2(x(t)-1)x(t)^{3}\\\ 0&0&0\\\
\end{array}\right),$ $\displaystyle A_{3}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}0&0&2(3-4x(t))x(t)^{2}&0\\\
0&0&0&6(x(t)-1)x(t)^{2}\\\ \end{array}\right),$ $\displaystyle A_{4}$
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccccc}0&0&12(1-2x(t))x(t)&0&0\\\
0&0&0&6x(t)(3x(t)-2)&0\\\ \end{array}\right),$ $\displaystyle A_{5}$
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccccc}0&0&12-48x(t)&0&0&0\\\
0&0&0&36x(t)-12&0&0\\\ \end{array}\right),$ $\displaystyle A_{6}$
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccccccc}0&0&-48&0&0&0&0\\\
0&0&0&36&0&0&0\\\ \end{array}\right),$ $\displaystyle A_{k}$ $\displaystyle=$
$\displaystyle 0_{2\times d_{2,k}},\quad k\geq 7.$

The fact that the ODE in (113) is so trivial to solve
($x\left({t}\right)=-t+C$) adds no simplicity to what follows below. The
fundamental matrix of $\mathrm{VE}_{1}$ and the first filter matrix are
equally simple:

$Y_{1}=\left(\begin{array}[]{cc}1&0\\\ 0&e^{x(0)-x(t)}\\\
\end{array}\right),\qquad F_{1}=\left(\begin{array}[]{cc}0&0\\\ 0&1\\\
\end{array}\right)$

and the generic form of the admissible solutions after applying Theorem 2.2.2
is the following: $f_{k}=\left({\frac{d}{dx}f_{k-1}\mid
g_{k}\left({x}\right)}\right)$ where $g_{k}$ is a solution to

$-(k-1)k(x-1)x^{3}g_{k-2}^{\prime}(x)-g_{k}^{\prime}(x)+(k-2)(k-1)k(x-1)x^{2}g_{k-2}(x)+kg_{k}(x)=0.$
(120)

and moreover $g_{k}\left({x}\right)=0$ if $k$ is even, and
$g_{1}\left({x}\right)=e^{x}$. Equation (120) has a solution (which of course
only merits being written if $k$ is odd)

$g_{k}\left({x}\right)=e^{kx}\int(k-1)k(\xi-1)\xi^{2}e^{-k\xi}\left({(k-2)g_{k-2}\left({\xi}\right)-\xi
g^{\prime}_{k-2}\left({\xi}\right)}\right)d\xi.$

The formal first integral of (112) takes the form

$F=\sum_{k=1}^{\infty}\frac{1}{k!}g_{k}\left({x}\right)y^{k}=\sum_{i=0}^{\infty}\left({-1}\right)^{i}e^{\left({2i+1}\right)x}y^{2i+1}G_{2i+1};$

the first terms of $G_{\star}$ are easy to discern,

$\displaystyle G_{1}$ $\displaystyle=$ $\displaystyle 1,$ $\displaystyle
G_{3}$ $\displaystyle=$
$\displaystyle\int_{x_{0}}^{x}e^{-2\xi}(\xi-1)^{2}\xi^{2}d\xi,$ $\displaystyle
G_{5}$ $\displaystyle=$
$\displaystyle\int_{x_{0}}^{x}\left({\xi-1}\right)^{3}e^{-4\xi}\xi^{5}d\xi+3\int_{\tau_{0}}^{\tau}\left({\xi-1}\right)^{2}e^{-2\xi}\xi^{2}\left(\int\left({\eta-1}\right)^{2}e^{-2\eta}\eta^{2}d\eta\right)d\xi$

and the general term is

$G_{2i+1}=\sum_{m=1}^{i}\sum_{j=1}^{\binom{i-1}{m-1}}D_{i-1,m-1,j}\Phi\left({\left({i-m}\right)\mathbf{C}_{m,i-m-1,j}+\mathbf{1}_{m},2\left({i-m}\right)\mathbf{C}_{m,i-m-1,j}+\mathbf{1}_{m},3\left({i-m}\right)\mathbf{C}_{m,i-m-1,j}+\mathbf{1}_{m}}\right)$

where we define the following constant vectors:
$\mathbf{1}_{n}:=\left({1,\dots,1}\right)\in{\mathbb{Z}}^{n}$, and
$C_{\star,\star,\star}$ are the columns of the symmetric product

${\mathrm{Id}}_{k}\odot\mathbf{1}^{T}_{d_{k,l}}=\left({C_{k,l,1}\mid
C_{k,l,2}\mid\cdots\mid
C_{k,l,d_{k,l+1}}}\right)\in{\mathrm{Mat}}_{k,d_{k,l+1}}\left({{\mathbb{Z}}}\right);$

(obviously $C_{k,l,j}=0$ for $l<0$), $\Phi$ is defined recursively as follows:
for every $\mathbf{a},\mathbf{b},\mathbf{c}\in{\mathbb{Z}}^{m}$,

$\Phi\left({\mathbf{a},\mathbf{b},\mathbf{c}}\right):=\left\\{\begin{array}[]{ll}\int_{x_{0}}^{x}\left({\tau-1}\right)^{a_{1}+1}e^{-\left({b_{1}+1}\right)\tau}\tau^{c_{1}}d\tau,&m=1,\\\
\int_{x_{0}}^{x}\left({\tau-1}\right)^{a_{1}+1}e^{-\left({b_{1}+1}\right)\tau}\tau^{c_{1}}\Phi\left({\left({a_{2},\dots,a_{m}}\right),\left({b_{2},\dots,b_{m}}\right),\left({c_{2},\dots,c_{m}}\right)}\right)d\tau,&m>1\end{array}\right.$

and $D_{n,m,j}$ is the $j^{\mathrm{th}}$ term in the vector
$\mathbf{{D}}_{n,m}$ obtained from discarding terms containing repeating
factors from the reverse-order version of
$\mathbf{{i}}_{n}^{T}\stackrel{{\scriptstyle
m}}{{\odot\cdots\odot}}\mathbf{{i}}_{n}^{T}$ and $\mathbf{{i}}_{n}$ is the
reverse-order sequence of the first $n$ odd integers $>1$, e.g.

$\mathbf{{D}}_{3,2}=\left({3\cdot 5,\;3\cdot 7,\;5\cdot
7}\right)=\left({D_{3,2,j}}\right)_{j=1,2,3,4},\quad\mathbf{{D}}_{4,3}=\left({3\cdot
5\cdot 7,\;3\cdot 5\cdot 9,\;3\cdot 7\cdot 9,\;5\cdot 7\cdot 9}\right)$

##  4 Further comments

Ever since it first took form in [3], the use of linearized higher variational
equations to characterize, obstruct or redefine integrability is proving
itself to be a welcome addition to the domain of dynamical systems. This
contribution arrives at a time in which other attempts (e.g. [24, 25]) at
finding methods to _integrate_ dynamical systems rather than proving them non-
integrable are gradually gaining traction.

A number of facts are germane to further studies:

  1. (i)

the potential shown by higher variational equations in finding adequate
changes of variables that will simplify the system into tractability and,
potentially, integrability (this is most exemplified in the sample systems
tackled in §3);

  2. (ii)

linked to the above, the possible application of a _Baker-Campbell-Hausdorff_
[12] sorts of formula upon variable transformations, using the decomposition
of a transformed fundamental matrix into a product of matrices to our
advantage;

  3. (iii)

the possible amenability of results in §2 above to machine learning [26];

  4. (iv)

the applicability of this automatic quadrature-algebraic method to the
application of generalized, multivariate _Padé approximants_ [9, 13] to first
integrals, which will be tackled in an immediate future;

  5. (v)

the fact that the autonomous system (1) need not be Hamiltonian, thus allowing
any arbitrary system to be considered and losing the serious constraint posed
by symplectic transformations, since variable transformations can now be
chosen freely;

  6. (vi)

also related to (v): the fact that this overrides the obstacles posed by the
direct study of variational systems, which only has an implementation to
Hamiltonian systems and requires first integrals to be meromorphic.

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