New Separation of Variables for the Classical XXX and XXZ Heisenberg Spin
Chains

New Separation of Variables for the Classical XXX
and XXZ Heisenberg Spin Chains

Guido MAGNANO † and Taras SKRYPNYK ${}^{\dagger}{}^{\ddagger}$

G. Magnano and T. Skrypnyk

† Università degli Studi di Torino, via Carlo Alberto 10, 10123, Torino,
Italia<EMAIL_ADDRESS><EMAIL_ADDRESS>
‡ Bogolyubov Institute for Theoretical Physics, Kyiv, Ukraine

Received December 20, 2019, in final form May 16, 2020; Published online June
02, 2020

We propose a non-standard separation of variables for the classical integrable
XXX and XXZ spin chains with degenerate twist matrix. We show that for the
case of such twist matrices one can interchange the role of classical
separating functions $A(u)$ and $B(u)$ and construct a new full set of
separated variables, satisfying simpler equation of separation and simpler
Abel equations in comparison with the standard separated variables of
Sklyanin. We show that for certain cases of the twist matrices the constructed
separated variables can be directly identified with action-angle coordinates.

integrable spin chains; quadratic Sklyanin brackets; separation of variables

37J35; 17B80

##  1 Introduction

Completely integrable Hamiltonian systems admitting Lax representation [12]
have been an object of constant interest in physics and mathematics during the
last forty years.

An important problem in the theory of integrable systems still to be solved in
general is the problem of variable separation. The separated variables
$x_{i}$, $p_{j}$, $i,j=1,\dots,N$ are a set of (quasi)canonical coordinates
such that the following system of equations is satisfied [24]

$\displaystyle\Phi_{i}(x_{i},p_{i},I_{1},\dots,I_{N},C_{1},\dots,C_{r})=0,\qquad
i=1,\dots,N,$

where $\Phi_{i}$ are certain functions, $I_{k}$ are Poisson-commuting
integrals of motion, $C_{i}$ are Casimir functions and $N$ is half of the
dimension of the phase space.

The separated coordinates (whenever they exist) allow one to write Abel-type
equations (see Section 2.1) which, in their turn, provide a possibility to
solve explicitly the Hamilton equations of motion upon resolving the
corresponding Abel–Jacobi inversion problem. Separated variables are also
important when solving quantum integrable models [24]. That is why the
construction of variable separation is a central issue in the theory of both
classical and quantum integrable systems.

In order to construct separated variables in many cases one can use a pair of
_separating functions_ $A(u)$, $B(u)$, which depend on the dynamical variables
and on a complex parameter $u$. The coordinates $x_{1},\dots,x_{N}$ are
determined as zeroes of $B(u)$,

$\displaystyle B(x_{i})=0,\qquad i=1,\dots,N,$ (1.1)

and (quasi)momenta are obtained as values of $A(u)$ in these zeros

$\displaystyle p_{i}=A(x_{i}),\qquad i=1,\dots,N.$ (1.2)

In the so-called Lax-integrable case (i.e., when the Hamilton equations of
motion can be written in Lax form), a prescription to obtain separating
functions $B(u)$, $A(u)$ for $\mathfrak{gl}(2)$-valued Lax matrices
$L(u)=\sum\limits_{i,j=1}^{2}L_{ij}(u)X_{ij}$, where $X_{ij}$, $i,j=1,2$ is a
standard basis of $\mathfrak{gl}(2)$
$(X_{ij})_{\alpha\beta}=\delta_{i\alpha}\delta_{j,\beta}$, have been
introduced in [21, 22, 24]111Often in the literature the choice
$B(u)=L_{12}(u)$ is made. Nevertheless, since for the Cartan-invariant
$r$-matrices the choice between $L_{12}(u)$ or $L_{21}(u)$ is arbitrary, we
prefer the above-made choice.

$\displaystyle B(u)=L_{21}(u),\qquad A(u)=L_{11}(u).$ (1.3)

The equations of separation in this case coincide with the spectral curve of
the Lax matrix

$\displaystyle\Phi_{i}(x_{i},p_{i},I_{1},\dots,I_{N},C_{1},\dots,C_{r})=\det(L(x_{i})-p_{i}\,{\rm
Id})=0,\qquad i=1,\dots,N.$

The constructed variables $x_{i}$, $p_{i}$ should be (quasi)canonical in order
for the theory to work

$\displaystyle\\{x_{i},p_{j}\\}=\delta_{ij}f_{i}(x_{i},p_{i}),\qquad\\{x_{i},x_{j}\\}=0,\qquad\\{p_{i},p_{j}\\}=0,\qquad\forall\,i,j=1,\dots,N.$

The separating functions $A(u)$ and $B(u)$ have to satisfy an appropriate
Poisson algebra [5, 6] in order to produce (quasi)canonical coordinates (see
Section 2.2 below). The right-hand side of this algebra, as well as the very
definition (1.1)–(1.2), is asymmetric in the functions $A(u)$, $B(u)$.
Nevertheless, in some cases the algebra of separating functions is symmetric
in $A(u)$ and $B(u)$ and has the following form

$\displaystyle\\{B(u),B(v)\\}=b(u,v)A(v)B(u)-b(v,u)A(u)B(v),$ (1.4a)
$\displaystyle\\{A(u),B(v)\\}=\alpha(u,v)A(v)B(u)-\beta(u,v)A(u)B(v),$ (1.4b)
$\displaystyle\\{A(u),A(v)\\}=a(u,v)A(v)B(u)-a(v,u)A(u)B(v).$ (1.4c)

This situation occurs when the Lax matrix satisfy quadratic tensor brackets
[7, 19]

$\displaystyle\\{{L}(u)\otimes 1,1\otimes{L}(v)\\}=[r(u,v),{L}(u)\otimes
L(v)].$ (1.5)

Here $r(u,v)=\sum\limits_{i,j,k,l=1}^{2}r_{ij,kl}(u,v)X_{ij}\otimes X_{kl}$,
$X_{ij}$, is a skew-symmetric classical $r$-matrix.

The symmetry of the separating algebra (1.4) poses a natural question: is it
possible (when the separating algebra is symmetric) to exchange the roles of
the separating functions? That is, is it possible to define new separated
variables as follows

$\displaystyle A(x_{i})=0,\qquad p_{i}=B(x_{i}),\qquad i=1,\dots,N?$ (1.6)

The answer to this question is not obvious. Indeed, while the reversed
definition (1.6) in the symmetric case guarantees the (quasi)canonicity of the
constructed separated coordinates, it does not guarantee the existence of the
equations of separation for them.222See Remark 4.3 for another – “banal”
explanation of the reversed definition (1.6) which is not used here.

In the present paper we are going to answer two general questions:

  1. 1.

For what $\mathfrak{gl}(2)\otimes\mathfrak{gl}(2)$-valued $r$-matrices the
separating algebra of functions $A(u)$ and $B(u)$ defined by (1.3) is
symmetric in $A(u)$ and $B(v)$, i.e., has the form (1.4)?

  2. 2.

When does the reversed definition (1.6) produce separated variables for such
$r$-matrices? That is, when do the corresponding quasi-canonical coordinates
satisfy equations of separation with the initial algebra of first integrals?

For the convenience of the reader we formulate our answers already in the
Introduction. In particular, the answer to the first question is contained in
the following proposition.

######  Proposition 1.1.

The functions $A(u)$ and $B(u)$ defined by (1.3) satisfy the algebra (1.4)
with respect to the brackets (1.5) if the components of the $r$-matrix satisfy
the following conditions

$\displaystyle r_{21,21}(u,v)=0,\qquad r_{21,11}(u,v)=0,\qquad
r_{11,21}(u,v)=0,$ (1.7a) $\displaystyle r_{12,12}(u,v)=0,\qquad
r_{12,22}(u,v)=0,\qquad r_{22,12}(u,v)=0,$ (1.7b) $\displaystyle
r_{22,22}(u,v)=r_{11,11}(u,v).$ (1.7c)

There are at least two $\mathfrak{gl}(2)\otimes\mathfrak{gl}(2)$ valued
classical skew-symmetric $r$-matrices satisfying the condition (1.7): the
standard rational and standard trigonometric $r$-matrices. For the case of
these $r$-matrices we proceed with the answer to the second question. For this
purpose we define also the class of the Lax matrices under the consideration.
We consider the most physically important case of the Lax matrices of the XXX
and XXZ models of $N$ classical spins with a twisted periodic boundary
conditions, i.e., the Lax matrices of the following form

$\displaystyle L(u)=L^{(\nu_{1})}(u)\cdots L^{(\nu_{N})}(u)C,$ (1.8)

where $C=\sum\limits_{i,j=1}^{2}c_{ij}X_{ij}$ is a two by two twist matrix
satisfying the following condition

$\displaystyle[r(u,v),C\otimes C]=0$ (1.9)

and $L^{(\nu_{i})}(u)$ is the Lax matrix with a simple pole in the point
$u=\nu_{i}$, corresponding to $i$-th cite of the classical spin chain, where
the classical spins satisfy the Poisson brackets of $\mathfrak{gl}(2)^{\oplus
N}$ (in the case of the rational $r$-matrix) and of the direct sum of $N$
trigonometric Sklyanin-type algebras (in the case of trigonometric
$r$-matrix). Observe that for the case of all the considered quadratic Poisson
algebras the non-trivial integrals of motion are generated by
$I(u)=\operatorname{tr}L(u)$.

The following theorem holds true:

######  Theorem 1.2.

Let the coordinates $x_{i}$, $p_{i}$, $i=1,\dots,N$ be defined by (1.6), the
functions $A(u)$ and $B(u)$ be defined by (1.3) and the Lax matrix $L(u)$ be
defined by (1.8). Let the matrix $C$ be such that $c_{11}\neq 0$. Then the
coordinates $x_{i}$, $p_{i}$ are separated coordinates for the classical
integrable system with the algebra of integrals of motion generated by
$I(u)=\operatorname{tr}L(u)$ if and only if

$\displaystyle\det C=0.$

For the rational XXX case the matrix $C=\sum\limits_{i,j=1}^{2}c_{ij}X_{ij}$
is an arbitrary constant degenerated matrix. In this case the case the
equations of separation are written as follows

$\displaystyle c_{12}p_{i}=c_{11}I(x_{i}),\qquad i=1,\dots,N,$ (1.10)

and we distinguish to cases: $c_{12}\neq 0$ and $c_{12}=0$. In the first case
the curve of separation is a rational one and the corresponding Abel equations
are the following

$\displaystyle\sum\limits_{i=1}^{N}\dfrac{c_{11}x_{i}^{N-j}}{c_{12}p_{i}}\frac{{\rm
d}x_{i}}{{\rm d}t_{k}}=-\delta^{j}_{k},\qquad j,k=1,\dots,N,$

where $\frac{{\rm d}x_{i}}{{\rm d}t_{k}}=\\{I_{k},x_{i}\\}$, $k=1,\dots,N$
etc. are easily integrated in terms of elementary functions. This is a
consequence of the fact that the separation curve (1.10) is a rational one.

The second case $c_{12}=0$ is even more special. In this case we also have
that $c_{22}=0$ and the equations of separation coincides with separating
polynomial and acquires the form

$\displaystyle I(x_{i})=A(x_{i})=0,\qquad i=1,\dots,N.$ (1.11)

The coordinates $x_{i}$ in this case become functions of the integrals of
motion and may be identified with action variables. The Abel equations are
written for the (quasi)conjugate variables $p_{i}$

$\displaystyle\sum\limits_{i=1}^{N}\dfrac{x_{i}^{N-j}}{\partial_{x_{i}}I(x_{i})}\frac{1}{p_{i}}\frac{{\rm
d}p_{i}}{{\rm d}t_{k}}=\delta^{j}_{k},\qquad j,k=1,\dots,N,$

and produce linear differential equations for the corresponding angle
coordinates $\phi_{i}=\ln p_{i}$.

In the trigonometric XXZ case the constant twist matrix $C$ is diagonal,
$C=\sum\limits_{i=1}^{2}c_{ii}X_{ii}$ and $c_{22}=0$ in the degenerated case.
That is why in this case there is only a special degenerated case of non-
standard separation of variables characterized by the equation of separation
(1.11), where the coordinates of separation coincide with action variables and
the canonically conjugated variables coincide with the angles of the Liouville
theorem.

To the best of our knowledge these are the first examples (at least in the
Lax-integrable case) when the variables of separation coincide with the
action-angle variables and their construction provides immediate solution of
the equations of motion without performing of a (generally speaking difficult)
task of solving of the Abel–Jacobi inversion problem.

At the end of the introduction let us make several bibliographical comments.
The Lax-pair based approach to the variable separation in its general form was
proposed by Sklyanin in [24] as a development of his previous idea [20, 21,
22]. In the classical case the idea of the approach may be traced further back
to the papers [2, 25]. In the quantum case the approach has obtained a lot of
attention in the literature, let us mention only the series of recent papers
[10, 11, 14, 17]. In the classical case, unfortunately, there have been very
few works on the subject. For the classical XXX model we can mention several
papers [4, 8, 18, 22, 23, 24]. For the classical XXZ model we can mention only
two papers on the subject [21] and [6]. To fill this gap in the knowledge and
to study the corresponding classical models in more details is one of the aims
of our paper.

The structure of the present paper is the following: in Section 2 we remind
general notions of the classical variable separation theory, in Section 3 we
consider Lax-integrable case, in Sections 4 and 5 we concentrate on the
examples of the classical XXX and XXZ models. In these sections we also
consider $N=2$ examples, investigating the corresponding cases in details. In
particular, we explicitly find the reconstruction formulae for them,
expressing the initial dynamical variables via the constructed coordinates of
separation and the values of the Casimir functions. At last, in Section 5 we
conclude and discuss the open problems.

##  2 Separation of variables

###  2.1 Definitions and notations

Let us recall the definitions of Liouville integrability and separation of
variables in the general theory of Hamiltonian systems. An integrable
Hamiltonian system with $N$ degrees of freedom is determined on a
$2N$-dimensional symplectic manifold $\mathcal{M}$ (symplectic leaf in
$(\mathcal{P},\\{\ ,\ \\})$ and $N$ independent functions (first integrals)
$I_{j}$ commuting with respect to the Poisson bracket

$\displaystyle\\{I_{i},I_{j}\\}=0,\qquad i,j=1,\dots,N$

(for the Hamiltonian $H$ of the system may be taken any first integral
$I_{j}$).

To find separated variables means to find (at least locally) a set of
coordinates $x_{i}$, $p_{j}$, $i,j=1,\dots,N$ such that there exist $N$
relations

$\displaystyle\Phi_{i}(x_{i},p_{i},I_{1},\dots,I_{N},C_{1},\dots,C_{r})=0,\qquad
i=1,\dots,N,$

where $C_{i}$, $i=1,\dots,r$ are Casimir functions and the coordinates
$x_{i}$, $p_{j}$, $i,j=1,\dots,N$ are canonical

$\displaystyle\\{x_{i},p_{j}\\}=\delta_{ij},\qquad\\{x_{i},x_{j}\\}=0,\qquad\\{p_{i},p_{j}\\}=0,\qquad\forall\,i,j=1,\dots,N.$

In the present paper it will be convenient for us to work with quasi-canonical
coordinates satisfying the following Poisson brackets

$\displaystyle\\{x_{i},p_{j}\\}=\delta_{ij}p_{j},\qquad\\{x_{i},x_{j}\\}=0,\qquad\\{p_{i},p_{j}\\}=0,\qquad\forall\,i,j=1,\dots,N.$

Clearly the variables $x_{i}$, $\phi_{j}=\log p_{j}$ will be canonical then.

It is possible to show that the coordinates of separation $x_{i}$ satisfy the
Abel-type equations

$\displaystyle\sum\limits_{i=1}^{N}\dfrac{\dfrac{\partial\Phi_{i}(x_{i},p_{i},I_{1},\dots,I_{N},C_{1},\dots,C_{r})}{\partial
I_{k}}}{\dfrac{\partial\Phi_{i}(x_{i},p_{i};I_{1},\dots,I_{N},C_{1},\dots,C_{r})}{\partial
p_{i}}}\frac{1}{p_{i}}\frac{{\rm d}x_{i}}{{\rm
d}t_{j}}=-\delta_{kj},\qquad\forall\,k,j=1,\dots,N,$ (2.1)

and similar Abel-type equations are satisfied by the momenta of separation
$p_{i}$

$\displaystyle\sum\limits_{i=1}^{N}\dfrac{\dfrac{\partial\Phi_{i}(x_{i},p_{i},I_{1},\dots,I_{N},C_{1},\dots,C_{r})}{\partial
I_{k}}}{\dfrac{\partial\Phi_{i}(x_{i},p_{i};I_{1},\dots,I_{N},C_{1},\dots,C_{r})}{\partial
x_{i}}}\frac{1}{p_{i}}\frac{{\rm d}p_{i}}{{\rm
d}t_{j}}=\delta_{kj},\qquad\forall\,k,j=1,\dots,N.$ (2.2)

These equations are the last step before the integration of the classical
equations of motion.

###  2.2 The method of separating functions

Let $B(u)$ and $A(u)$ be some functions of the dynamical variables and of an
auxiliary complex parameter $u$, which is constant with respect to the bracket
$\\{\ ,\ \\}$. Let the points $x_{i}$, $i=1,\dots,N$ be zeros of the function
$B(u)$ and $p_{i}$, $i=1,\dots,N$ be the values of $A(u)$ in these points. We
wish to obtain (quasi)canonical Poisson brackets among these new coordinates
using the Poisson brackets between functions $B(u)$ and $A(u)$. The following
proposition holds true [5].

######  Proposition 2.1.

Let the coordinates $x_{i}$ and $p_{j}$, $i,j=1,\dots,p$ be defined as
$B(x_{i})=0$, $p_{j}=A(x_{j})$. Let the functions $A(u)$, $B(u)$ satisfy the
following Poisson algebra

$\displaystyle\\{B(u),B(v)\\}=b(u,v)B(u)-b(v,u)B(v),$ (2.3a)
$\displaystyle\\{A(u),B(v)\\}=\alpha(u,v)B(u)-\beta(u,v)B(v),$ (2.3b)
$\displaystyle\\{A(u),A(v)\\}=a(u,v)B(u)-a(v,u)B(v).$ (2.3c)

Then the Poisson bracket between the functions $x_{i}$ and $p_{j}$,
$\forall\,i,j=1,\dots,N$ are the following

$\displaystyle\\{x_{i},x_{j}\\}=0,\qquad\forall\,i,j=1,\dots,N,$
$\displaystyle\\{x_{j},p_{i}\\}=0,\qquad\text{if}\quad i\neq j,$
$\displaystyle\\{p_{i},p_{j}\\}=0,\qquad\forall\,i,j=1,\dots,N.$

If, moreover also the condition

$\displaystyle\lim\limits_{u\rightarrow
v}(\alpha(u,v)B(u)-\beta(u,v)B(v))=A(v)\partial_{v}B(v)+\gamma(v)B(v)$

holds, then the corresponding Poisson brackets are quasi-canonical, i.e.,

$\displaystyle\\{x_{i},p_{i}\\}=p_{i},\qquad\forall\,i=1,\dots,N.$

######  Remark 2.2.

Observe that the coefficients $a(u,v)$, $b(u,v)$, $\alpha(u,v)$, $\beta(u,v)$,
$\gamma(v)$ above may depend not only on the spectral parameters but also on
the dynamical variables.

######  Remark 2.3.

Observe that in general separating algebra (2.3) is asymmetric in the
functions $A(u)$, $B(v)$. Nevertheless for some dynamical coefficients
$a(u,v)$, $b(u,v)$, $\alpha(u,v)$, $\beta(u,v)$ it may become symmetric in the
functions $A(u)$, $B(v)$. In such a case the functions $A(u)$, $B(v)$ become
interchangeable and one can “invert” the procedure, defining separated
coordinates also as follows: $A(x_{i})=0$, $p_{j}=B(x_{j})$. This is the
situation that will be studied in this article.

##  3 Separation of variables: Lax-integrable case

###  3.1 The equations of separation

Let us specify the above theory, i.e., equations of separation and separating
functions for the Lax-integrable case, when Hamiltonian equations of motion
with respect to a Hamiltonian $H$ can be written in Lax form [12] with a
spectral-parameter-dependent Lax matrix

$\displaystyle\dot{L}(u)=\left[L(u),M_{H}(u)\right]$

According to the “magic recipe” of Sklyanin in this case the role of all
equations of separation is played by a single equation, namely the spectral
curve of the Lax matrix

$\displaystyle\Phi_{i}(x_{i},p_{i},I_{1},\dots,I_{N},C_{1},\dots,C_{r})=\det(L(x_{i})-p_{i}\,{\rm
Id})=0,\qquad i=1,\dots,N.$

This hypothesis works good for the case of the $\mathfrak{gl}(n)$-valued Lax
matrices [1, 4, 8, 18, 24]. In what follows we will consider the simplest case
of the $\mathfrak{gl}(2)$-valued Lax matrices.

###  3.2 The separating functions

Let $X_{ij}$, $i,j=1,2$ be a standard basis in $\mathfrak{gl}(2)$ with the
commutation relations

$\displaystyle[X_{ij},X_{kl}]=\delta_{kj}X_{il}-\delta_{il}X_{kj}.$

The $\mathfrak{gl}(2)$-valued Lax matrix is written as follows

$\displaystyle L(u)=\sum\limits_{i,j=1}^{2}L_{ij}(u)X_{ij}.$

Following the “magic recipe” in its standard version [24] we will assume that
the separating functions $A(u)$ and $B(u)$ are defined as follows:

$\displaystyle A(u)=L_{11}(u),\qquad B(u)=L_{21}(u).$ (3.1)

###  3.3 The separating algebra and its symmetries

Now we will require that the algebra of the functions $A(u)$ and $B(u)$
defined by (3.1) have the particular form (2.3). For this purpose it is
necessary at first to define the Poisson brackets among the components of the
Lax matrix.

In this paper we will consider the case of the so-called quadratic Sklyanin
bracket [19]

$\displaystyle\\{{L}(u)\otimes
1,1\otimes{L}(v)\\}=\big{[}r^{12}(u,v),{L}(u)\otimes L(v)\big{]},$ (3.2)

where

$\displaystyle
r^{12}(u,v)=\sum\limits_{i,j,k,l=1}^{2}r_{ij,kl}(u,v)X_{ij}\otimes X_{kl}$
(3.3)

is a skew-symmetric classical $r$-matrix: $r^{12}(u,v)=-r^{21}(v,u)$ (see [3,
7, 15, 19]).

The algebra (2.3) is satisfied by the above functions $A(u)$ and $B(u)$ under
certain conditions on the $r$-matrix. In more detail, the following
proposition holds true.333See [5, 6] for the generalization of this
proposition onto the case of $\mathfrak{gl}(n)$-valued Lax matrices.

######  Proposition 3.1.

The functions $A(u)$ and $B(u)$ defined by (3.1) satisfy the algebra (2.3)
with respect to the brackets (3.2) if the components of the $r$-matrix satisfy
the following conditions

$\displaystyle r_{21,21}(u,v)=0,\qquad r_{21,11}(u,v)=0,\qquad
r_{11,21}(u,v)=0.$ (3.4)

###### Proof.

The proof of the proposition is achieved by direct calculation.∎

Observe that the algebra (2.3) is very asymmetric in the functions $A(u)$ and
$B(u)$. It is asymmetric also in the considered case after imposing the
conditions (3.4). We wish to investigate the question when the quadratic
Poisson algebra of the functions $A(u)$ and $B(u)$ not only satisfies (2.3)
but also has its right-hand side to be symmetric in functions $A(u)$ and
$B(u)$. Evidently, such symmetry will require more rigid conditions on the
$r$-matrix.

The following proposition holds true.

######  Proposition 3.2.

  1. $(i)$

The functions $A(u)$ and $B(u)$ defined by (3.1) satisfy the algebra (2.3)
with respect to the brackets (3.2) in symmetric way with respect to $A(u)$ and
$B(v)$ if the components of the $r$-matrix satisfy the following conditions

$\displaystyle r_{21,21}(u,v)=0,\qquad r_{21,11}(u,v)=0,\qquad
r_{11,21}(u,v)=0,$ (3.5a) $\displaystyle r_{12,12}(u,v)=0,\qquad
r_{12,22}(u,v)=0,\qquad r_{22,12}(u,v)=0,$ (3.5b) $\displaystyle
r_{22,22}(u,v)=r_{11,11}(u,v).$ (3.5c)

  2. $(ii)$

Under the conditions (3.5) the algebra of separating functions reads as
follows

$\displaystyle\\{B(u),B(v)\\}=r_{21,22}(u,v)A(u)B(v)+r_{22,21}(u,v)B(u)A(v),$
(3.6a)
$\displaystyle\\{A(u),B(v)\\}=(r_{11,22}(u,v)-r_{11,11}(u,v))B(v)A(u)+r_{12,21}(u,v)B(u)A(v),$
(3.6b)
$\displaystyle\\{A(u),A(v)\\}=r_{11,12}(u,v)A(u)B(v)+r_{12,11}(u,v)B(u)A(v).$
(3.6c)

###### Proof.

The proposition is proven by direct calculation.∎

######  Remark 3.3.

There are at least two $\mathfrak{gl}(2)\otimes\mathfrak{gl}(2)$ valued
classical skew-symmetric $r$-matrices satisfying the condition (3.5): standard
rational and standard trigonometric $r$-matrices.

Using the symmetry of separating algebra, one can invert the procedure
described in the previous subsection and define the (quasi)canonical
coordinates as follows

$\displaystyle A(x_{i})=0,\qquad p_{i}=B(x_{i}).$ (3.7)

In the next section we will address, for the cases of rational and
trigonometric $r$-matrices, the question whether the canonical coordinates
defined by (3.7) are separation coordinates.

##  4 Classical XXX spin model

###  4.1 Poisson brackets and Lax matrix

Let us now consider the simplest possible case of the standard rational
$r$-matrix

$\displaystyle r(u,v)=\frac{\sum\limits_{i,j=1}^{2}X_{ij}\otimes X_{ji}}{u-v}$
(4.1)

and describe the corresponding Lax matrices $L(u)$, satisfying the quadratic
brackets (3.2).

As it is well-known, the Lax matrices of the spin-chain models satisfying the
quadratic brackets (3.2) can be written in the following product form

$\displaystyle L^{(1,2,\dots,N)}(u)=L^{(\nu_{1})}(u)\cdots L^{(\nu_{N})}(u),$
(4.2)

where $N$ is arbitrary and the basic one-spin matrices $L^{(i)}(u)$ are
written as follows

$\displaystyle
L^{(\nu_{k})}(u)=1+\frac{1}{u-\nu_{k}}\sum\limits_{i,j=1}^{2}S^{(k)}_{ji}X_{ij}.$

Here $\nu_{i}\neq\nu_{j}$, when $i\neq j$, $i,j=1,\dots,N$ and the Poisson
brackets among the coordinates $S^{(n)}_{ij}$, $S^{(m)}_{kl}$ are those of the
direct sum $\mathfrak{gl}(2)^{\oplus N}$

$\displaystyle\big{\\{}{S}_{ij}^{(m)},{S}_{kl}^{(n)}\big{\\}}=\delta^{mn}\big{(}\delta_{kj}{S}_{il}^{(m)}-\delta_{il}{S}_{kj}^{(m)}\big{)}.$

Hereafter we will be interested in the Lax matrices of the following form

$\displaystyle L(u)=L^{(1,2,\dots,N)}(u)C,$ (4.3)

where $C$ is an arbitrary constant matrix
$C=\sum\limits_{i,j=1}^{2}c_{ij}X_{ij}$. By the virtue of the fact that in the
case of the $r$-matrix (4.1)

$\displaystyle[r(u,v),C\otimes C]=0,$

the Lax matrix (4.3) satisfies the Poisson bracket (3.2) with a rational
$r$-matrix (4.1).

The Lax matrix (4.3) is a Lax matrix of the XXX type $\mathfrak{gl}(2)$
Heisenberg spin chain with twisted periodic boundary conditions defined by the
constant twist matrix $C$ [24].

###  4.2 Integrals and Casimir functions

Let us now consider the integrable system on $\mathfrak{gl}(2)^{\oplus N}$
defined with the help of the Lax matrix (4.3). Its mutually Poisson-commuting
integrals of motion are constructed from the characteristic polynomial of the
Lax matrix $L(u)$

$\displaystyle I(w,u)=\det(L(u)-w\,{\rm Id})=w^{2}-w\operatorname{tr}L(u)+\det
L(u).$

The function $\det L(u)=\det C\det L^{(1,2,\dots,N)}(u)$ is a generating
function of the Casimir functions of the quadratic brackets (3.2). The
corresponding Casimir functions it contains consist of different combinations
of the linear and quadratic Casimir functions of $\mathfrak{gl}(2)^{\oplus N}$

$\displaystyle c_{k}=S^{(k)}_{11}+S^{(k)}_{22},\qquad
C_{k}=S^{(k)}_{11}S^{(k)}_{22}-S^{(k)}_{12}S^{(k)}_{21}.$

The generating function of integrals of motion is

$\displaystyle I(u)=\operatorname{tr}L(u).$

More explicitly, we have that

$\displaystyle
I(u)=\frac{1}{\prod\limits_{i=1}^{N}(\nu_{i}-u)}\left((-1)^{N}(c_{11}+c_{22})u^{N}+\sum\limits_{k=1}^{N}I_{k}u^{N-k}\right),$

where the Hamiltonians $I_{k}$ are non-homogeneous polynomials of the degree
up to $k$ in the dynamical variables. In particular, we have that

$\displaystyle
I_{1}=-\left(\sum\limits_{i,j=1}^{2}c_{ij}\sum\limits_{k=1}^{N}S^{(k)}_{ij}+\sum\limits_{l=1}^{N}\nu_{l}\sum\limits_{i=1}^{2}c_{ii}\right),$
$\displaystyle
I_{2}=\sum\limits_{i,j=1}^{2}c_{ij}\sum\limits_{l,m=1,\,l<m}^{N}\sum\limits_{k=1}^{2}S^{(m)}_{ik}S^{(l)}_{kj}+\sum\limits_{i,j=1}^{2}c_{ij}\sum\limits_{k=1}^{N}\left(\sum\limits_{l=1}^{N}\nu_{l}-\nu_{k}\right)S^{(k)}_{ij}+\sum\limits_{k,l=1,\,k<l}^{N}\nu_{l}\nu_{k}\sum\limits_{i=1}^{2}c_{ii}.$

The integral $I_{2}$ may be viewed as a quadratic Hamiltonian of the classical
spin chain of $N$ spins. It evidently does not have the Heisenberg-type form.
The Heisenberg-type Hamiltonians are obtained from $I(u)$ (in the quantum case
it works in the representation of $\mathfrak{gl}(2)^{\oplus N}$ algebra by
Pauli matrices) using formulae of the following type [24]

$\displaystyle H_{n}=\left(\frac{{\rm d}^{n}\ln I(u)}{{\rm
d}u^{n}}\right)\bigg{|}_{u=0}.$

We will not consider or use the Hamiltonians $H_{n}$ here: they do not enter
directly into the spectral curve nor into the equation of separation, and
therefore are not useful for our purposes.

######  Remark 4.1.

We have preserved seemingly non-important constant terms in the integrals of
motion because they make non-trivial contributions to the equations of
separation.

###  4.3 Non-standard variable separation: general degenerated case

####  4.3.1 Separated variables and equation of separation

The rational $r$-matrix (4.1) evidently satisfies the conditions (3.5). Hence,
using our inverted definition (3.7) one can define the (quasi)canonical
coordinates as follows

$\displaystyle A(x_{i})=0,\qquad p_{i}=B(x_{i}),$

where the functions $A(u)$ and $B(u)$ are standardly defined using the Lax
matrix $L(u)$

$\displaystyle A(u)=L_{11}(u),\qquad B(u)=L_{21}(u).$

For the Lax operator defined by the formulae (4.2), (4.3) we obtain

$\displaystyle
A(u)=\frac{1}{\prod\limits_{i=1}^{N}(\nu_{i}-u)}\left((-1)^{N}c_{11}u^{N}+\sum\limits_{k=1}^{N}A_{k}u^{N-k}\right),$
$\displaystyle
B(u)=\frac{1}{\prod\limits_{i=1}^{N}(\nu_{i}-u)}\left((-1)^{N}c_{21}u^{N}+\sum\limits_{k=1}^{N}B_{k}u^{N-k}\right),$

where $A_{k}$, $B_{k}$ are the functions of the dynamical variables
$S^{(l)}_{ij}$ and the constants $c_{ij}$. From their explicit form it follows
that in the case $c_{11}\neq 0$ one can take $A(u)$ for the role of separating
polynomial: it produces $N$ separated coordinates $x_{i}$.

The following theorem holds true.

######  Theorem 4.2.

Let us assume that the matrix $C$ is such that $c_{11}\neq 0$.444In the case
$c_{11}=0$, $c_{22}\neq 0$ one should redefine the separating functions:
$A(u)=L_{22}(u)$, $B(u)=L_{12}(u).$ Let the coordinates $x_{i}$, $p_{i}$,
$i=1,\dots,N$ be defined by (3.7), where $A(u)=L_{11}(u)$, $B(u)=L_{21}(u)$
and the Lax matrix $L(u)$ be defined by the formulae (4.2), (4.3). Then the
coordinates $x_{i}$, $p_{i}$ are the separated coordinates for the integrable
system with the algebra of integrals of motion generated by
$I(u)=\operatorname{tr}L(u)$ if and only if

$\displaystyle\det C\equiv c_{11}c_{22}-c_{12}c_{21}=0.$

The equations of separation have in this case the following form

$\displaystyle c_{12}p_{i}=c_{11}I(x_{i}),\qquad i=1,\dots,N.$ (4.4)

######  Remark 4.3.

One may wonder if the exchange of roles between $A(u)$ and $B(u)$ is not
trivially equivalent to swapping the Lax matrix elements $L_{11}(u)$ and
$L_{21}(u)$ by a mere change of Lax representation. However, this is not so:
indeed, one might take as a new Lax matrix the product $L^{C_{0}}\equiv
C_{0}L(u)$, where $C_{0}=(X_{12}+X_{21})$.555In the quantum case a similar
trick was used in the paper [17]. In this case we have

$\displaystyle B^{C_{0}}(u)\equiv L^{C_{0}}_{21}(u)=L_{11}(u)=A(u),\qquad
A^{C_{0}}(u)\equiv L^{C_{0}}_{11}(u)=L_{21}(u)=B(u).$

But in this way the integrals of motion are also changed — instead of the
generating function

$\displaystyle I(u)=\operatorname{tr}L(u)=L_{11}(u)+L_{22}(u)$

one obtains a new generating function

$\displaystyle
I^{C_{0}}(u)=\operatorname{tr}L^{C_{0}}(u)=L_{12}(u)+L_{21}(u).$

After the change, the Sklyanin recipe yields separation variables for
_another_ integrable system with a different algebra of first integrals, and
the equations of separation coincide with the spectral curve of the Lax matrix
$L^{C_{0}}(u)$: no progress is obtained for the original Lax equation. We
stress that this is totally different from the method that we are introducing
here, where the inversion between $A(u)$ and $B(u)$ produces a different set
of separated variables for the same integrable system.

###### Proof of Theorem 4.2.

First of all let us observe, that by the virtue of the relation (3.6) and the
explicit form of the $r$-matrix we obtain

$\displaystyle\\{B(u),B(v)\\}=0,$
$\displaystyle\\{A(u),B(v)\\}=\frac{B(u)A(v)-B(v)A(u)}{u-v},$
$\displaystyle\\{A(u),A(v)\\}=0.$

From this and the Proposition 2.1 (with the reversed role of the function
$A(u)$ and $B(u)$) it immediately follows that the coordinates $x_{i}$,
$p_{i}$ defined as above are (quasi)canonical

$\displaystyle\\{x_{i},x_{j}\\}=0,\qquad\\{p_{i},p_{j}\\}=0,\qquad\\{x_{i},p_{i}\\}=p_{i},\qquad\forall\,i=1,\dots,N.$

Now it is left to show that they satisfy the equations of separation. We will
prove them for any Lax matrix of the form

$\displaystyle L(u)=\tilde{L}(u)C,$

where $\tilde{L}(u)$ is any Lax matrix satisfying quadratic brackets (3.2) and
$C$ is a degenerated constant matrix satisfying (1.9). Taking into account the
explicit form of the functions $B(u)$ and $A(u)$ in terms of the components of
the Lax matrix $\tilde{L}(u)$ and the matrix $C$ we obtain

$\displaystyle B(u)=\tilde{L}_{21}(u)c_{11}+\tilde{L}_{22}(u)c_{21},\qquad
A(u)=\tilde{L}_{11}(u)c_{11}+\tilde{L}_{12}(u)c_{21},$ $\displaystyle
I(u)=\tilde{L}_{11}(u)c_{11}+\tilde{L}_{12}(u)c_{21}+\tilde{L}_{21}(u)c_{12}+\tilde{L}_{22}(u)c_{22}.$

From this it immediately follows that the following identity holds true

$\displaystyle
I(u)=\frac{c_{12}}{c_{11}}B(u)+A(u)+\frac{c_{11}c_{22}-c_{12}c_{21}}{c_{11}}\tilde{L}_{22}(u).$
(4.5)

Using this we obtain that in the case when $c_{11}c_{22}-c_{12}c_{21}=0$ the
(quasi)canonical coordinates $x_{i}$, $p_{i}$ satisfy the following equations
of separation

$\displaystyle I(x_{i})=\frac{c_{12}}{c_{11}}p_{i},\qquad i=1,\dots,N.$

This proves the sufficiency of the conditions $c_{11}c_{22}-c_{12}c_{21}=0$
for separability with our “inversed” separating functions $A(u)$ and $B(u)$.

The necessity follows from the results of [13] that provide necessary and
sufficient differential (with respect to the time flows) conditions for the
coefficients of the separating polynomial ($A(u)$ in our case) to generate
separated coordinates. Making use of these conditions and calculating them
explicitly for our system we obtain that they are satisfied if and only if
$c_{11}c_{22}-c_{12}c_{21}=0$. ∎

######  Remark 4.4.

Observe that in the case $c_{21}\neq 0$ one can use more standard scheme
taking the function $B(u)$ to be separating polynomial. In the case $c_{21}=0$
the choice of $B(u)$ to be separating polynomial is a bad choice, due to the
fact that $B(u)$ in this case has less roots than necessary. If we also have
that $c_{22}=0$, $c_{11}\neq 0$, then by the virtue of the Theorem above the
good choice of the separating polynomial is the function $A(u)$.

######  Remark 4.5.

Observe that the equation of separation $I(x_{i})=\frac{c_{12}}{c_{11}}p_{i}$
can be also obtained from the spectral curve of the Lax matrix $L(u)$. Indeed,
putting $\det(C)=0$ we obtain that the spectral curve degenerates to the form

$\displaystyle w(w-I(u))=0.$

If we now put

$\displaystyle u=x_{i},\qquad w=\dfrac{c_{12}}{c_{11}}p_{i}$

we obtain the equation (4.4).

######  Remark 4.6.

Observe also that if we had chosen the coordinates as follows:
$p_{i}=A(x_{i})$, $B(x_{i})=0$ under the condition $\det(C)=0$ the equation of
separation would have been $I(x_{i})=p_{i}$, not $c_{12}p_{i}=c_{11}I(x_{i})$
which makes a difference, especially in the case $c_{12}=0$.

####  4.3.2 The Abel equations

The important step in the theory of classical variable separation is re-
writing of the equations of motion in the Abel form. This permits one to
integrate the equations of motion for $x_{i}$ resolving the so-called
Abel–Jacobi inversion problem.

The following proposition holds true.

######  Proposition 4.7.

The equations of motion for the coordinates of separation $x_{i}$ described in
the previous subsection are written in the Abel-type form as follows

$\displaystyle\sum\limits_{i=1}^{N}\dfrac{x_{i}^{N-j}}{(-1)^{N}(c_{11}+c_{22})x_{i}^{N}+\sum\limits_{k=1}^{N}I_{k}x_{i}^{N-k}}\frac{{\rm
d}x_{i}}{{\rm d}t_{k}}=-\delta^{j}_{k},\qquad j,k=1,\dots,N,$ (4.6)

where $\frac{{\rm d}x_{i}}{{\rm d}t_{k}}=\\{I_{k},x_{i}\\}$, $k=1,\dots,N$,
etc.

###### Idea of the proof.

The Abel equations (4.6) are the consequences of the general Abel-type
equations (2.1) and are calculated using the explicit form of the equations of
separation in the considered case. ∎

######  Remark 4.8.

Let us remark that the equations (4.6) are easily integrated and the
corresponding Abel–Jacobi problem is resolved in terms of the elementary
functions.

###  4.4 Non-standard variable separation: special degenerated case

####  4.4.1 Separated variables and equation of separation

Let us now consider special degenerated case $c_{12}=0$, $c_{22}=0$. Although
this case is a partial case of the above construction, but it is so special
that deserves a separate consideration.

The following corollary of the Theorem 4.2 holds true.

######  Corollary 4.9.

Let us assume that the matrix $C$ is such that $c_{11}\neq 0$,
$c_{12}=c_{22}=0$. Let the coordinates $x_{i}$, $p_{i}$, $i=1,\dots,N$ be
defined by (3.7), where $A(u)=L_{11}(u)$, $B(u)=L_{21}(u)$ and the Lax matrix
$L(u)$ be defined by the formulae (4.2), (4.3). Then the equations of
separation have the following form

$\displaystyle A(x_{i})=I(x_{i})=0,\qquad i=1,\dots,N.$

###### Proof.

At first we observe that in the case $c_{12}=c_{22}=0$ we have that
$\det(C)=0$. As it follows from the equations (4.4) in the case $c_{12}=0$ the
equation of separation are written simply as follows

$\displaystyle I(x_{i})=0,\qquad i=1,\dots,N.$

But, taking into account that $c_{22}=0$ we will obtain from (4.5)

$\displaystyle I(u)=A(u).$

The corollary is proven.∎

######  Remark 4.10.

Observe that in the considered special degenerated case the coordinates
$x_{i}$ are the action variables and are expressed via the integrals of
motion.

####  4.4.2 The Abel equations

Let us now consider the Abel-type equations. In this case, due to the fact
that the coordinates $x_{i}$ are expressed via the integrals, the Abel-type
equations should be written for momenta.

The following proposition holds true.

######  Proposition 4.11.

The equations of motion for the momenta of separation $p_{i}$ described in the
previous subsection are written in the Abel-type form as follows

$\displaystyle\sum\limits_{i=1}^{N}\dfrac{x_{i}^{n-j}}{\sum\limits_{l=1}^{N-1}(N-l)x_{i}^{N-l-1}I_{l}}\frac{1}{p_{i}}\frac{{\rm
d}p_{i}}{{\rm d}t_{k}}=\delta^{j}_{k},\qquad j,k=1,\dots,N,$ (4.7)

where $\frac{{\rm d}p_{i}}{{\rm d}t_{k}}=\\{I_{k},x_{i}\\}$, $k=1,\dots,N$ and
$x_{i}$ are expressed in terms of the integrals of motion as solutions of the
equations of separation $I(x_{i})=0$, $i=1,\dots,N$.

###### Idea of the proof.

The Abel equations (4.7) are the consequences of the general Abel-type
equations (2.2) and are calculated using the explicit form of the equations of
separation in the considered case. ∎

Introducing the angle variables $\phi_{i}$ by

$\displaystyle p_{i}=\exp{\phi_{i}},\qquad i=1,\dots,N$

we obtain that the equations (4.7) are easily integrated as follows

$\displaystyle\sum\limits_{i=1}^{N}\dfrac{x_{i}^{N-j}}{\sum\limits_{l=1}^{N-1}(N-l)x_{i}^{N-l-1}I_{l}}\big{(}\phi_{i}-\phi_{i}^{0}\big{)}=\big{(}t_{j}-t_{j}^{0}\big{)},\qquad
j=1,\dots,N.$

The last equations have the following explicit solution

$\displaystyle\phi_{i}=\phi_{i}^{0}+\sum\limits_{j=1}\big{(}M(I)^{-1}\big{)}_{ij}\big{(}t_{j}-t_{j}^{0}\big{)},\qquad\text{where}\qquad
M(I)_{ij}=\frac{x_{i}^{N-j}}{\sum\limits_{l=1}^{N-1}(N-l)x_{i}^{N-l-1}I_{l}},$

and we remind that $x_{i}$ are functions of the integrals of motion.

To summarize: the case $c_{12}=c_{22}=0$ admits immediate explicit
construction of the action-angle variables in the corresponding XXX spin
chain.

###  4.5 $\boldsymbol{N=2}$ example

####  4.5.1 The model

Let us consider the simplest non-trivial example of $N=2$. We will have

$\displaystyle L^{(1,2)}(u)=L^{(\nu_{1})}(u)L^{(\nu_{2})}(u),$

and the basic one-spin matrices $L^{(i)}(u)$ are written as follows

$\displaystyle
L^{(\nu_{k})}(u)=1+\frac{1}{u-\nu_{k}}\sum\limits_{i,j=1}^{2}S^{(k)}_{ji}X_{ij}.$

Hereafter for the purpose of convenience we will locate the poles as follows:
$\nu_{1}=1$, $\nu_{2}=-1$, and set $S^{(1)}_{ij}=S_{ij}$,
$S^{(2)}_{kl}=T_{kl}$. The Poisson brackets are those of
$\mathfrak{gl}(2)\oplus\mathfrak{gl}(2)$

$\displaystyle\\{{S}_{ij},{S}_{kl}\\}=(\delta_{kj}{S}_{il}-\delta_{il}{S}_{kj}),$
(4.8a)
$\displaystyle\\{{T}_{ij},{T}_{kl}\\}=(\delta_{kj}{T}_{il}-\delta_{il}{T}_{kj}),$
(4.8b) $\displaystyle\\{{T}_{ij},{S}_{kl}\\}=0.$ (4.8c)

As previously we will be interested in the Lax matrices of the following form:

$\displaystyle L(u)=L^{(1,2}(u)C,$

where the matrix $C$ is an arbitrary constant degenerated matrix
$C=\sum\limits_{i,j=1}^{2}c_{ij}X_{ij}$, $\det(C)=0$.

The generating function of integrals of motion is

$\displaystyle I(u)=\operatorname{tr}L(u).$

More explicitly, we will have that

$\displaystyle
I(u)=\frac{1}{(1-u^{2})}\big{(}(c_{11}+c_{22})u^{2}+I_{1}u+I_{2}\big{)},$

where the Hamiltonians $I_{k}$ are non-homogeneous polynomials in the
dynamical variables

$\displaystyle I_{1}=-\sum\limits_{i,j=1}^{2}c_{ij}(S_{ij}+T_{ij}),$
$\displaystyle
I_{2}=\sum\limits_{i,j=1}^{2}c_{ij}\sum\limits_{k=1}^{2}T_{ik}S_{kj}-\sum\limits_{i,j=1}^{2}c_{ij}(S_{ij}-T_{ij})-\sum\limits_{i=1}^{2}c_{ii}.$

####  4.5.2 Separated variables

Let, as previously, the functions $A(u)$ and $B(u)$ are defined as follows

$\displaystyle A(u)=L_{11}(u),\qquad B(u)=L_{21}(u).$

For the considered $N=2$ case of the Lax matrix we obtain

$\displaystyle
A(u)=\frac{1}{(1-u^{2})}\big{(}c_{11}u^{2}-((S_{11}+T_{11})c_{11}+(T_{21}+S_{21})c_{21})u$
$\displaystyle\hphantom{A(u)=}{}+(T_{11}-S_{11}+S_{11}T_{11}+S_{21}T_{12}-1)c_{11}+(T_{21}-S_{21}+S_{21}T_{22}+T_{21}S_{11})c_{21}\big{)},$
$\displaystyle
B(u)=\frac{1}{(1-u^{2})}\big{(}c_{21}u^{2}-(c_{11}(T_{12}+S_{12})-c_{21}(T_{22}+S_{22}))u$
$\displaystyle\hphantom{B(u)=}{}+(S_{12}T_{11}+T_{12}S_{22}+T_{12}-S_{12}+)c_{11}+(S_{12}T_{21}-1+S_{22}T_{22}+T_{22}-S_{22})c_{21}\big{)}.$

We will again define the (quasi)canonical coordinates as follows

$\displaystyle A(x_{i})=0,\qquad p_{i}=B(x_{i}),$

that is $x_{1}$, $x_{2}$ are the solutions of the following quadratic equation

$\displaystyle c_{11}x^{2}-((S_{11}+T_{11})c_{11}+(T_{21}+S_{21})c_{21})x$
$\displaystyle\qquad{}+(T_{11}-S_{11}+S_{11}T_{11}+S_{21}T_{12}-1)c_{11}+(T_{21}-S_{21}+S_{21}T_{22}+T_{21}S_{11})c_{21}=0$

and

$\displaystyle
p_{i}=\frac{1}{(1-x_{i}^{2})}\big{(}c_{21}x_{i}^{2}-(c_{11}(T_{12}+S_{12})-c_{21}(T_{22}+S_{22}))x_{i}$
$\displaystyle\hphantom{p_{i}=}{}+(S_{12}T_{11}+T_{12}S_{22}+T_{12}-S_{12}+)c_{11}+(S_{12}T_{21}-1+S_{22}T_{22}+T_{22}-S_{22})c_{21}\big{)}.$

By virtue of our general theory the coordinates $x_{i}$, $p_{i}$ are quasi-
canonical

$\displaystyle\\{x_{i},x_{j}\\}=0,\qquad\\{p_{i},p_{j}\\}=0,\qquad\\{x_{i},p_{i}\\}=p_{i},\qquad\forall\,i,j=1,2,$
(4.9)

and satisfy the following equation of separation

$\displaystyle\frac{c_{12}}{c_{11}}p_{i}=\frac{1}{(1-x_{i}^{2})}\big{(}(c_{11}+c_{22})x_{i}^{2}+I_{1}x_{i}+I_{2}\big{)},\qquad
i=1,2.$

In the case $c_{12}=0$, $c_{22}=0$ these equations of separation degenerate to
the form

$\displaystyle\big{(}c_{11}x_{i}^{2}+I_{1}x_{i}+I_{2}\big{)}=0,\qquad i=1,2,$
(4.10)

i.e., the coordinates of separation $x_{i}$ became the action coordinates in
this case.

####  4.5.3 The reconstruction formulae

In order to completely resolve the problem in the case $N=2$ it is necessary
to be able to reconstruct the original dynamical variables from the separated
variables and Casimir functions, i.e., it is necessary to solve the following
system of eight linear-quadratic equations

$\displaystyle
c_{11}x_{i}^{2}+((-S_{11}-T_{11})c_{11}-(S_{21}+T_{21})c_{21})x_{i}+(-1-S_{11}+T_{11}+S_{11}T_{11}+S_{21}T_{12})c_{11}$
$\displaystyle\qquad{}+(T_{21}+S_{21}T_{22}+T_{21}S_{11}-S_{21})c_{21}=0,\qquad
i=1,2,$
$\displaystyle\big{(}x_{i}^{2}-1\big{)}p_{i}=(c_{21}x_{i}^{2}+((-T_{12}-S_{12})c_{11}+(-T_{22}-S_{22})c_{21})x_{i}$
$\displaystyle\qquad{}+(-S_{12}+T_{12}S_{22}+S_{12}T_{11}+T_{12})c_{11}$
$\displaystyle\qquad{}+(T_{22}-S_{22}+S_{22}T_{22}+S_{12}T_{21}-1)c_{21}),\qquad
i=1,2,$ $\displaystyle C_{1}=S_{11}S_{22}-S_{12}S_{21},\qquad
C_{2}=T_{11}T_{22}-T_{12}T_{21},\qquad c_{1}=S_{11}+S_{22},\qquad
c_{2}=T_{11}+T_{22}$

on the eight variables $S_{ij}$, $T_{ij}$, $i,j=1,2$.

Hereafter for the purpose of simplicity we will put $c_{21}=0$, $c_{22}=0$ and
$c_{1}=c_{2}=0$.

The following theorem holds true.

######  Theorem 4.12.

  1. $(i)$

The mechanical coordinates $S_{ij}$, $T_{ij}$ are parametrized by the
separated coordinates $x_{i}$, $p_{i}$ and the values of the Casimir functions
$C_{1}$, $C_{2}$ as follows

$\displaystyle
S_{11}=\bigl{(}((x_{1}+1)(x_{1}-1)^{2}p_{1}-(1+x_{2})(-1+x_{2})^{2}p_{2})C_{1}$
$\displaystyle\hphantom{S_{11}=}{}+(-1+x_{2})^{2}(x_{1}+1)(x_{1}-1)^{2}p_{1}-(1+x_{2})(-1+x_{2})^{2}(x_{1}-1)^{2}p_{2}\bigr{)}$
$\displaystyle\hphantom{S_{11}=}{}\times\bigl{(}((x_{1}-1)(x_{1}+1)p_{1}-(-1+x_{2})(1+x_{2})p_{2})C_{1}$
$\displaystyle\hphantom{S_{11}=}{}+(-1+x_{2})^{2}(x_{1}-1)(x_{1}+1)p_{1}-(-1+x_{2})(1+x_{2})(x_{1}-1)^{2}p_{2}\bigr{)}^{-1},$
(4.11a) $\displaystyle
S_{12}=c_{11}\bigl{(}(x_{1}-x_{2})C_{1}^{2}+(x_{1}-x_{2})(x_{1}^{2}-2x_{1}+x_{2}^{2}+2-2x_{2})C_{1}$
$\displaystyle\hphantom{S_{12}=}{}+(x_{2}-1)^{2}(x_{1}-1)^{2}(x_{1}-x_{2})\bigr{)}\bigl{(}((x_{1}-1)(x_{1}+1)p_{1}-(x_{2}-1)(1+x_{2})p_{2})C_{1}$
$\displaystyle\hphantom{S_{12}=}{}+(x_{2}-1)^{2}(x_{1}-1)(x_{1}+1)p_{1}-(x_{2}-1)(1+x_{2})(x_{1}-1)^{2}p_{2}\bigr{)}^{-1},$
(4.11b) $\displaystyle S_{22}=-S_{11},\qquad
S_{21}=\frac{(S_{11}S_{22}-C_{1})}{S_{12}},$ (4.11c) $\displaystyle
T_{11}=\bigl{(}((x_{1}-1)(x_{1}+1)(1+x_{2})p_{1}-(x_{2}-1)(1+x_{2})(x_{1}+1)p_{2})C_{1}$
$\displaystyle\hphantom{T_{11}=}{}+(1+x_{2})(x_{2}-1)^{2}(x_{1}-1)(x_{1}+1)p_{1}-(x_{2}-1)(1+x_{2})(x_{1}+1)(x_{1}-1)^{2}p_{2}\bigr{)}$
$\displaystyle\hphantom{T_{11}=}{}\times\bigl{(}((x_{1}-1)(x_{1}+1)p_{1}-(x_{1}-1)(1+x_{2})p_{2})C_{1}$
$\displaystyle\hphantom{T_{11}=}{}+(x_{2}-1)^{2}(x_{1}-1)(x_{1}+1)p_{1}-(x_{2}-1)(1+x_{2})(x_{1}-1)^{2}p_{2}\bigr{)}^{-1},$
(4.11d) $\displaystyle
T_{12}=\frac{p_{1}p_{2}}{c_{11}}(x_{2}-1)(1+x_{2})(x_{1}-1)(x_{1}+1)(x_{1}-x_{2})\bigl{(}((x_{1}-1)(x_{1}+1)p_{1}$
$\displaystyle\hphantom{T_{12}=}{}-(x_{2}-1)(1+x_{2})p_{2})C_{1}+(x_{2}-1)^{2}(x_{1}-1)(x_{1}+1)p_{1}$
$\displaystyle\hphantom{T_{12}=}{}-(x_{2}-1)(1+x_{2})(x_{1}-1)^{2}p_{2})\bigr{)}^{-1},$
(4.11e) $\displaystyle T_{22}=-T_{11},\qquad
T_{21}=\frac{(T_{11}T_{22}-C_{2})}{T_{12}}.$ (4.11f)

  2. $(ii)$

If the Poisson relations among $p_{i}$ and $q_{j}$ are quasi-canonical (4.9)
then the variables $S_{ij}$, $T_{ij}$ given by (4.11) satisfy the Poisson
brackets (4.8).

###### Idea of the proof.

Item (i) of the theorem is proven by direct calculations. The item (ii)
follows from our general theory and can be also checked by direct
calculations. ∎

######  Remark 4.13.

Using the formulae (4.11) one can obtain the expression for the integrals
$I_{1}$ and $I_{2}$ in terms of the separated variables

$\displaystyle
I_{1}=\frac{c_{12}(x_{1}-1)(x_{1}+1)p_{1}}{c_{11}(x_{1}-x_{2})}-\frac{c_{12}(x_{2}-1)(1+x_{2})p_{2}}{c_{11}(x_{1}-x_{2})}-(x_{1}+x_{2})c_{11},$
(4.12a) $\displaystyle
I_{2}=\frac{-c_{12}x_{2}(x_{1}-1)(x_{1}+1)p_{1}}{c_{11}(x_{1}-x_{2})}+\frac{c_{12}x_{1}(x_{2}-1)(1+x_{2})p_{2}}{c_{11}(x_{1}-x_{2})}+x_{1}x_{2}c_{11}.$
(4.12b)

####  4.5.4 Abel-type equations: general degenerated case

Using either our general theory or the explicit form of the Hamiltonians in
terms of the separated variables (4.12) one easily derives the Abel-type
equations. In case $c_{12}\neq 0$ the Abel equations are written for the
separated coordinates $x_{i}$ as follows

$\displaystyle\frac{x_{1}}{(c_{11}+c_{22})x_{1}^{2}+I_{1}x_{1}+I_{2}}\frac{{\rm
d}x_{1}}{{\rm
d}t_{1}}+\frac{x_{2}}{(c_{11}+c_{22})x_{2}^{2}+I_{1}x_{2}+I_{2}}\frac{{\rm
d}x_{2}}{{\rm d}t_{1}}=1,$
$\displaystyle\frac{x_{1}}{(c_{11}+c_{22})x_{1}^{2}+I_{1}x_{1}+I_{2}}\frac{{\rm
d}x_{1}}{{\rm
d}t_{2}}+\frac{x_{2}}{(c_{11}+c_{22})x_{2}^{2}+I_{1}x_{2}+I_{2}}\frac{{\rm
d}x_{2}}{{\rm d}t_{2}}=0,$
$\displaystyle\frac{1}{(c_{11}+c_{22})x_{1}^{2}+I_{1}x_{1}+I_{2}}\frac{{\rm
d}x_{1}}{{\rm
d}t_{1}}+\frac{1}{(c_{11}+c_{22})x_{2}^{2}+I_{1}x_{2}+I_{2}}\frac{{\rm
d}x_{2}}{{\rm d}t_{1}}=0,$
$\displaystyle\frac{1}{(c_{11}+c_{22})x_{1}^{2}+I_{1}x_{1}+I_{2}}\frac{{\rm
d}x_{1}}{{\rm
d}t_{2}}+\frac{1}{(c_{11}+c_{22})x_{2}^{2}+I_{1}x_{2}+I_{2}}\frac{{\rm
d}x_{2}}{{\rm d}t_{2}}=1.$

They are easily resolved in terms of elementary functions.

####  4.5.5 Abel-type equations: special degenerated case

In case $c_{12}=0$, $c_{22}=0$, $c_{11}\neq 0$ the Abel equations written for
the separated coordinates $x_{i}$ are trivial due to the fact that $x_{i}$ in
this case are expressed via the integrals of motion. The non-trivial Abel-type
equations are written for the conjugated momenta $p_{i}$

$\displaystyle\frac{x_{1}}{(x_{1}-x_{2})p_{1}}\frac{{\rm d}p_{1}}{{\rm
d}t_{1}}+\frac{x_{2}}{(x_{2}-x_{1})p_{2}}\frac{{\rm d}p_{2}}{{\rm
d}t_{1}}=c_{11},$ $\displaystyle\frac{1}{(x_{1}-x_{2})p_{1}}\frac{{\rm
d}p_{1}}{{\rm d}t_{1}}+\frac{1}{(x_{2}-x_{1})p_{2}}\frac{{\rm d}p_{2}}{{\rm
d}t_{1}}=0,$ $\displaystyle\frac{x_{1}}{(x_{1}-x_{2})p_{1}}\frac{{\rm
d}p_{1}}{{\rm d}t_{2}}+\frac{x_{2}}{(x_{2}-x_{1})p_{2}}\frac{{\rm
d}p_{2}}{{\rm d}t_{2}}=0,$
$\displaystyle\frac{1}{(x_{1}-x_{2})p_{1}}\frac{{\rm d}p_{1}}{{\rm
d}t_{2}}+\frac{1}{(x_{2}-x_{1})p_{2}}\frac{{\rm d}p_{2}}{{\rm
d}t_{2}}=c_{11},$

where we have used the following relations among $x_{i}$ and $I_{i}$

$\displaystyle I_{1}=-c_{11}(x_{1}+x_{2}),\qquad I_{2}=c_{11}x_{1}x_{2}.$

Introducing the angle variables $p_{i}=\exp{\phi_{i}}$, $i=1,2$ we obtain that

$\displaystyle\phi_{1}=\phi_{1}^{0}+c_{11}\big{(}\big{(}t_{1}-t_{1}^{0}\big{)}-x_{2}\big{(}t_{2}-t_{2}^{0}\big{)}\big{)},\qquad\phi_{2}=\phi_{2}^{0}+c_{11}\big{(}\big{(}t_{1}-t_{1}^{0}\big{)}-x_{1}\big{(}t_{2}-t_{2}^{0}\big{)}\big{)},$

where $x_{i}$, $i=1,2$ are the constants of motion calculated from the
equations (4.10).

##  5 Classical XXZ spin model

###  5.1 The $\boldsymbol{r}$-matrix, the basic Lax matrix and quadratic
brackets

Let us consider the following $r$-matrix

$\displaystyle
r^{12}(u_{1},u_{2})=\frac{1}{2}\frac{u_{1}+u_{2}}{u_{1}-u_{2}}(X_{11}\otimes
X_{11}+X_{22}\otimes X_{22})$
$\displaystyle\hphantom{r^{12}(u_{1},u_{2})=}{}+\frac{u_{1}}{u_{1}-u_{2}}X_{12}\otimes
X_{21}+\frac{u_{2}}{u_{1}-u_{2}}X_{21}\otimes X_{12}.$ (5.1)

It is the so-called standard trigonometric $r$-matrix [3] in the special gauge
[9]. We are interested in obtaining of the Lax matrices satisfying the algebra
(3.2) for this $r$-matrix.

The following proposition holds true [6].

######  Proposition 5.1.

Let $\nu\neq 0$, $\nu\neq\infty$. Then the Lax matrix $L^{(\nu)}(u)$ defined
by the formula

$\displaystyle
L^{(\nu)}(u)=\sum\limits_{i=1}^{2}S_{0i}X_{ii}+\dfrac{1}{2}\frac{\nu+u}{\nu-u}\sum\limits_{i=1}^{2}S_{ii}X_{ii}+\frac{u}{\nu-u}S_{21}X_{12}+\frac{\nu}{\nu-u}S_{12}X_{21},$
(5.2)

where the Poisson brackets among the coordinate functions $S_{0i}$, $S_{ii}$,
$S_{ij}$ are the following

$\displaystyle\\{S_{0i},S_{0j}\\}=0,\qquad\\{S_{0i},S_{jj}\\}=0,\qquad\\{S_{ii},S_{jj}\\}=0,\qquad
i,j=1,2,$ (5.3a)
$\displaystyle\\{S_{0i},S_{ij}\\}=\frac{1}{4}S_{ii}S_{ij},\qquad\\{S_{ii},S_{ij}\\}=S_{0i}S_{ij},\qquad
i,j=1,2,$ (5.3b)
$\displaystyle\\{S_{0i},S_{ji}\\}=-\frac{1}{4}S_{ii}S_{ji},\qquad\\{S_{ii},S_{ji}\\}=-S_{0i}S_{ji},\qquad
i,j=1,2,$ (5.3c)
$\displaystyle\\{S_{ij},S_{ji}\\}=S_{0j}S_{ii}-S_{0i}S_{jj},\qquad i,j=1,2$
(5.3d)

satisfies Poisson brackets (3.2).

Let us describe the central elements (Casimir functions) of the quadratic
Poisson algebra (5.3). As it is well-known (see, e.g., [16] for the detailed
proof), the Casimir functions of any quadratic Poisson algebra are obtained by
expanding in powers of $u$ of the following generating function

$\displaystyle C(u)=\det(L(u)).$

More explicitly, we will have the following expansion, yielding three Casimir
functions $C_{k}$

$\displaystyle C(u)=\frac{1}{4(\nu-u)^{2}}\sum\limits_{k=0}^{2}u^{k}C_{2-k},$

where

$\displaystyle C_{0}=\prod\limits_{i=1}^{2}(2S_{0i}-S_{ii}),\qquad
C_{1}=2(S_{11}S_{22}-2S_{12}S_{21}-4S_{01}S_{02}),\qquad
C_{2}=\prod\limits_{i=1}^{2}(2S_{0i}+S_{ii}).$

Besides, there also exist two additional quadratic Casimir functions

$\displaystyle c_{i}=4S_{0i}^{2}-S_{ii}^{2},\qquad i=1,2.$

The functions $c_{1}$, $c_{2}$, $C_{1}$, $C_{2}$ are functionally independent.
That is why the dimension of the generic symplectic leaf of the brackets (5.3)
is two.

######  Remark 5.2.

Observe, that there exists also the following quadratic Casimir function

$\displaystyle C_{2}^{\prime}=4(S_{01}-S_{02})^{2}-(S_{11}+S_{22})^{2}.$

On its zero level surface one can put

$\displaystyle S_{01}=S_{02},\qquad S_{11}=-S_{22}$

and arrive exactly to the trigonometric degeneration of the famous elliptic
Sklyanin algebra on the space $\mathfrak{sl}(2)$ extended with the help of
one-dimensional center $S_{0}=S_{01}=S_{02}$.

###  5.2  The Lax matrix and integrals of the classical XXZ model

Following the general theory of the quadratic $r$-matrix brackets it is
possible to consider the product of Lax matrices

$\displaystyle
L^{(1,2,\dots,N)}(u)={L}^{(\nu_{1})}(u)\cdots{L}^{(\nu_{N})}(u),$ (5.4)

where the elementary Lax matrices $L^{\nu_{k}}(u)$ are given by the formula
(5.2)

$\displaystyle
L^{(\nu_{k})}(u)=\sum\limits_{i=1}^{2}S^{(k)}_{0i}X_{ii}+\dfrac{1}{2}\frac{\nu_{k}+u}{\nu_{k}-u}\sum\limits_{i=1}^{2}S^{(k)}_{ii}X_{ii}+\frac{u}{\nu_{k}-u}S^{(k)}_{21}X_{12}+\frac{\nu_{k}}{\nu_{k}-u}S^{(k)}_{12}X_{21}.$

The Poisson brackets among the functions $S^{(k)}_{0i}$, $S^{(k)}_{ij}$
exactly repeat the Poisson brackets (5.3). The Poisson brackets among the
functions $S^{(m)}_{0i}$, $S^{(m)}_{ij}$ and $S^{(n)}_{0k}$, $S^{(n)}_{kl}$
are trivial

$\displaystyle\big{\\{}S^{(m)}_{0i},S^{(n)}_{0j}\big{\\}}=0,\qquad\big{\\{}S^{(m)}_{0i},S^{(n)}_{kl}\big{\\}}=0,\qquad\big{\\{}S^{(m)}_{ij},S^{(n)}_{kl}\big{\\}}=0,$
$\displaystyle\forall\,i,j,k,l=1,2,\qquad m,n=1,\dots,N,\qquad m\neq n.$

The independent Casimir functions are the functions

$\displaystyle
C^{(m)}_{1}=2\big{(}S^{(m)}_{11}S^{(m)}_{22}-2S^{(m)}_{12}S^{(m)}_{21}-4S^{(m)}_{01}S^{(m)}_{02}\big{)},$
$\displaystyle
C^{(m)}_{2}=\prod\limits_{i=1}^{2}\big{(}2S^{(m)}_{0i}+S^{(m)}_{ii}\big{)},\qquad
m=1,\dots,N.$

Besides there exist also the following quadratic Casimir functions

$\displaystyle
c^{(m)}_{i}=4\big{(}S^{(m)}_{0i}\big{)}^{2}-\big{(}S^{(m)}_{ii}\big{)}^{2},\qquad
i=1,2,\qquad m=1,\dots,N.$ (5.5)

The dimension of the generic symplectic leaf is $2N$. So we have to have $N$
independent integrals for the complete integrability of the Hamiltonian system
on the generic symplectic leaf. We will obtain them from the following Lax
matrix

$\displaystyle L(u)=L^{(1,2,\dots,N)}(u)C,$ (5.6)

where $C$ is a constant matrix, in a standard way

$\displaystyle I(u)=\operatorname{tr}(L(u)).$

The matrix $C$ is not arbitrary. It should satisfy the following condition

$\displaystyle[r(u,v),C\otimes C]=0.$ (5.7)

The following proposition is proven by the direct calculation.

######  Proposition 5.3.

The matrix $C=\sum\limits_{i,j=1}^{2}c_{ij}X_{ij}$ satisfy the condition (5.7)
for the case of the trigonometric $r$-matrix (5.1) if $C$ is diagonal, i.e.,
$c_{12}=c_{21}=0$.

The generating function of the integrals of motion is written in the case of
the diagonal matrices $C$ as follows

$\displaystyle
I(u)=\frac{1}{\prod\limits_{i=1}^{N}(u-\nu_{i})}\sum\limits_{k=0}^{N}u^{N-k}I_{k},$

where

$\displaystyle
I_{0}=c_{11}\prod\limits_{k=1}^{N}\big{(}2S^{(k)}_{01}-S^{(k)}_{11}\big{)}+c_{22}\prod\limits_{k=1}^{N}\big{(}2S^{(k)}_{02}-S^{(k)}_{22}\big{)},$
$\displaystyle
I_{1}=(-1)^{N}c_{11}\left(\sum\limits_{l=1}^{N}\big{(}2S^{(l)}_{01}-S^{(l)}_{11}\big{)}\prod\limits_{k=1,k\neq
l}^{N}\nu_{k}\big{(}2S^{(k)}_{01}+S^{(k)}_{11}\big{)}\right.$
$\displaystyle\left.\hphantom{I_{1}=}{}-\sum\limits_{l,m=1,l<m}^{N}4\nu_{m}S^{(m)}_{12}S^{(l)}_{21}\prod\limits_{k=1,k\neq
l,m}^{N}\nu_{k}\big{(}2S^{(k)}_{01}+S^{(k)}_{11}\big{)}\right)$
$\displaystyle\hphantom{I_{1}=}{}+(-1)^{N}c_{22}\left(\sum\limits_{l=1}^{N}\big{(}2S^{(l)}_{02}-S^{(l)}_{22}\big{)}\prod\limits_{k=1,k\neq
l}^{N}\nu_{k}\big{(}2S^{(k)}_{02}+S^{(k)}_{22}\big{)}\right.$
$\displaystyle\left.\hphantom{I_{1}=}{}-\sum\limits_{l,m=1,l<m}^{N}4\nu_{l}S^{(m)}_{21}S^{(l)}_{12}\prod\limits_{k=1,k\neq
l,m}^{N}\nu_{k}\big{(}2S^{(k)}_{02}+S^{(k)}_{22}\big{)}\right),$
$\displaystyle\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots$
$\displaystyle
I_{N}=(-1)^{N}\left(\prod\limits_{l=1}^{N}\nu_{l}\right)\left(c_{11}\prod\limits_{k=1}^{N}\big{(}2S^{(k)}_{01}+S^{(k)}_{11}\big{)}+c_{22}\prod\limits_{k=1}^{N}\big{(}2S^{(k)}_{02}+S^{(k)}_{22}\big{)}\right),$

etc. It is possible to show that the functions $I_{1},\dots,I_{N}$ are
functionally independent on the generic symplectic leaf, while the function
$I_{0}$ is dependent on the Casimir functions and other integrals. In
particular, in the relevant case $c_{22}=0$ we will have

$\displaystyle
I_{0}=\frac{(-1)^{N}c_{11}^{2}\prod\limits_{l=1}^{N}\nu_{l}c^{(1)}_{l}}{I_{N}},$
(5.8)

where $c^{(1)}_{l}$ are the above-defined Casimir functions (5.5).

###  5.3 Non-standard variable separation

####  5.3.1 Separated variables and equation of separation

The trigonometric $r$-matrix (5.1) evidently satisfies the conditions (3.5).
Hence, using our inverted definition (3.7) one can define the (quasi)canonical
coordinates as follows

$\displaystyle A(x_{i})=0,\qquad p_{i}=B(x_{i}),$

where the functions $A(u)$ and $B(u)$ are standardly defined using the Lax
matrix $L(u)$

$\displaystyle A(u)=L_{11}(u),\qquad B(u)=L_{21}(u).$

For the Lax operator defined by the formulae (5.4), (5.6) we also obtain

$\displaystyle
A(u)=\frac{1}{\prod\limits_{i=1}^{N}(u-\nu_{i})}\left(\sum\limits_{k=0}^{N}A_{k}u^{N-k}\right),\qquad
B(u)=\frac{1}{\prod\limits_{i=1}^{N}(u-\nu_{i})}\left(\sum\limits_{k=0}^{N}B_{k}u^{N-k}\right),$

where $A_{k}$, $B_{k}$ are the functions of the dynamical variables
$S^{(l)}_{0j}$, $S^{(l)}_{ij}$ and the constants $c_{ij}$.

In particular, we have that

$\displaystyle
A_{0}=c_{11}\prod\limits_{k=1}^{N}\big{(}2S^{(k)}_{01}-S^{(k)}_{11}\big{)},\qquad
A_{N}=(-1)^{N}c_{11}\left(\prod\limits_{l=1}^{N}\nu_{l}\right)\prod\limits_{k=1}^{N}\big{(}2S^{(k)}_{01}+S^{(k)}_{11}\big{)}.$

It is important that on a generic symplectic leaf one has $A_{0}\neq 0$,
$A_{N}\neq 0$, i.e., the separating polynomial $A(u)$ has degree $N$ and
provide the needed number of the coordinates of separation.

The following theorem holds true.

######  Theorem 5.4.

Let $C=\operatorname{diag}(c_{11},c_{22})$ and $c_{11}\neq 0$.666In the case
$c_{11}=0$, $c_{22}\neq 0$ one redefines the separating function putting
$A(u)=L_{22}(u)$, $B(u)=L_{12}(u).$ Let the coordinates $x_{i}$, $p_{i}$,
$i=1,\dots,N$ be defined by (3.7), where $A(u)=L_{11}(u)$, $B(u)=L_{21}(u)$
and the Lax matrix $L(u)$ be defined by the formulae (5.4), (5.6). Then the
coordinates $x_{i}$, $p_{i}$ are separated coordinates for the integrable
system with the algebra of integrals of motion generated by
$I(u)=\operatorname{tr}L(u)$ if and only if $c_{22}=0$. The equations of
separation in this case have the following form

$\displaystyle I(x_{i})=A(x_{i})=0,\qquad i=1,\dots,N,$

i.e., the coordinates $x_{i}$ coincide with action variables for the system.

###### Sketch of the proof.

Proof of the theorem is similar to the proof of the corresponding theorem in
the case of the rational $r$-matrix. The only difference is the algebra of
separating functions that reads in this case as follows

$\displaystyle\\{B(u),B(v)\\}=0,$
$\displaystyle\\{A(u),B(v)\\}=-\frac{1}{2}\frac{u+v}{u-v}A(u)B(v)+\frac{v}{u-v}A(v)B(u),$
$\displaystyle\\{A(u),A(v)\\}=0.$

By the virtue of this algebra we obtain in the trigonometric case the quasi-
canonical Poisson brackets of the following form

$\displaystyle\\{x_{i},x_{j}\\}=0,\qquad\\{p_{i},p_{j}\\}=0,\qquad\\{x_{i},p_{i}\\}=x_{i}p_{i},\qquad\forall\,i,j=1,2.$

The other specific feature of the trigonometric case is that we are always in
the special degeneration settings, i.e., $c_{12}=c_{21}=0$ and the equation of
separation is

$\displaystyle I(x_{i})=0,\qquad i=1,\dots,N.$

It holds true if and only if $\det(C)=c_{11}c_{22}=0$, i.e., $c_{22}=0$ for
non-trivial $c_{11}$. In this case $I(u)=A(u)$. This proves the theorem. ∎

####  5.3.2 The Abel equations

Let us now consider the Abel-type equations. In this case, due to the fact
that the coordinates $x_{i}$ are expressed via the integrals, the Abel-type
equations should be written for momenta.

Similar to the case of the rational $r$-matrix, the following proposition
holds true.

######  Proposition 5.5.

The equations of motion for the momenta of separation $p_{i}$ described in the
previous subsection are written in the Abel-type form as follows

$\displaystyle\sum\limits_{i=1}^{N}\left(\dfrac{x_{i}^{N-j}}{\sum\limits_{l=1}^{N-1}(N-l)x_{i}^{N-l-1}I_{l}}-\delta_{jN}\dfrac{x_{i}^{N}\dfrac{(-1)^{N}c_{11}^{2}\prod\limits_{l=1}^{N}\nu_{l}c^{(1)}_{l}}{I^{2}_{N}}}{\sum\limits_{l=1}^{N-1}(N-l)x_{i}^{N-l-1}I_{l}}\right)\frac{1}{x_{i}p_{i}}\frac{{\rm
d}p_{i}}{{\rm d}t_{k}}=\delta^{j}_{k},$ (5.9)

$j,k=1,\dots,N$, where $\frac{{\rm d}p_{i}}{{\rm d}t_{k}}=\\{I_{k},x_{i}\\}$,
$k=1,\dots,N$ and $x_{i}$ are expressed in terms of the integrals of motion as
solutions of the equations of separation $I(x_{i})=0$, $i=1,\dots,N$.

###### Proof.

The Abel equations (4.7) are the consequences of the general Abel-type
equations (2.2) and are calculated using the explicit form of the equations of
separation in the considered case. ∎

######  Remark 5.6.

The different form of the Abel equations (5.9) with respect to those in the
rational case is explained by the different form of the quasi-canonical
Poisson brackets and the fact that the integral $I_{0}$ in the pencil $I(u)$
is expressed via $I_{N}$ as in (5.8).

As in the rational case, introducing the angle variables

$\displaystyle p_{i}=\exp{\phi_{i}},\qquad i=1,\dots,N$

we obtain that the equations (5.9) are easily integrated as follows

$\displaystyle\sum\limits_{i=1}^{N}\left(\dfrac{x_{i}^{N-j}}{\sum\limits_{l=1}^{N-1}(N-l)x_{i}^{N-l-1}I_{l}}-\delta_{jN}\dfrac{x_{i}^{N}\dfrac{(-1)^{N}c_{11}^{2}\prod\limits_{l=1}^{N}\nu_{l}c^{(1)}_{l}}{I^{2}_{N}}}{\sum\limits_{l=1}^{N-1}(N-l)x_{i}^{N-l-1}I_{l}}\right)\frac{\big{(}\phi_{i}-\phi_{i}^{0}\big{)}}{x_{i}}=\big{(}t_{j}-t_{j}^{0}\big{)},$
(5.10)

$j=1,\dots,N$. The explicit solution of (5.10) is constructed in the same
manner as in the rational case.

###  5.4 $\boldsymbol{N=2}$ example

####  5.4.1 The model

Let us consider the simplest non-trivial example of $N=2$. We will have

$\displaystyle L^{(1,2)}(u)=L^{(\nu_{1})}(u)L^{(\nu_{2})}(u),$

and the basic one-spin matrices $L^{(\nu_{k})}(u)$ are written as follows

$\displaystyle
L^{(\nu_{k})}(u)=\sum\limits_{i=1}^{2}S^{(k)}_{0i}X_{ii}+\dfrac{1}{2}\frac{\nu_{k}+u}{\nu_{k}-u}\sum\limits_{i=1}^{2}S^{(k)}_{ii}X_{ii}$
$\displaystyle\hphantom{L^{(\nu_{k})}(u)=}{}+\frac{u}{\nu_{k}-u}S^{(k)}_{21}X_{12}+\frac{\nu_{k}}{\nu_{k}-u}S^{(k)}_{12}X_{21},\qquad
k=1,2.$

Hereafter for the purpose of convenience we will put $\nu_{1}=1$,
$\nu_{2}=-1$,

$\displaystyle S^{(1)}_{ij}=S_{ij},\qquad S^{(1)}_{0j}=S_{0j},\qquad
S^{(2)}_{kl}=T_{kl},\qquad S^{(2)}_{0j}=T_{0j}.$

The Poisson brackets are those of the direct sum of two quadratic algebras
(5.3)

$\displaystyle\\{S_{0i},S_{0j}\\}=0,\qquad\\{S_{0i},S_{jj}\\}=0,\qquad\\{S_{ii},S_{jj}\\}=0,\qquad
i,j=1,2,$ (5.11a)
$\displaystyle\\{S_{0i},S_{ij}\\}=\frac{1}{4}S_{ii}S_{ij},\qquad\\{S_{ii},S_{ij}\\}=S_{0i}S_{ij},\qquad
i,j=1,2,$ (5.11b)
$\displaystyle\\{S_{0i},S_{ji}\\}=-\frac{1}{4}S_{ii}S_{ji},\qquad\\{S_{ii},S_{ji}\\}=-S_{0i}S_{ji},\qquad
i,j=1,2,$ (5.11c)
$\displaystyle\\{S_{ij},S_{ji}\\}=S_{0j}S_{ii}-S_{0i}S_{jj},\qquad i,j=1,2,$
(5.11d)
$\displaystyle\\{T_{0i},T_{0j}\\}=0,\qquad\\{T_{0i},T_{jj}\\}=0,\qquad\\{T_{ii},T_{jj}\\}=0,\qquad
i,j=1,2,$ (5.11e)
$\displaystyle\\{T_{0i},T_{ij}\\}=\frac{1}{4}T_{ii}T_{ij},\qquad\\{T_{ii},T_{ij}\\}=T_{0i}T_{ij},\qquad
i,j=1,2,$ (5.11f)
$\displaystyle\\{T_{0i},T_{ji}\\}=-\frac{1}{4}T_{ii}T_{ji},\qquad\\{T_{ii},T_{ji}\\}=-T_{0i}T_{ji},\qquad
i,j=1,2,$ (5.11g)
$\displaystyle\\{T_{ij},T_{ji}\\}=T_{0j}T_{ii}-T_{0i}T_{jj},\qquad i,j=1,2,$
(5.11h)
$\displaystyle\\{S_{ij},T_{kl}\\}=\\{S_{0i},T_{0j}\\}=\\{S_{0i},T_{kl}\\}=\\{T_{0j},S_{kl}\\}=0.$
(5.11i)

The Casimir functions of the quadratic Sklyanin-type parenthesis of the direct
sum are

$\displaystyle C^{(1)}_{1}=2(S_{11}S_{22}-2S_{12}S_{21}-4S_{01}S_{02}),\qquad
C^{(2)}_{1}=2(T_{11}T_{22}-2T_{12}T_{21}-4T_{01}T_{02}),$ $\displaystyle
C^{(1)}_{2}=(2S_{01}+S_{11})(2S_{02}+S_{22}),\qquad
C^{(2)}_{2}=(2T_{01}+T_{11})(2T_{02}+T_{22}).$

Besides, there exist also the following quadratic Casimir functions

$\displaystyle c^{(1)}_{i}=4(S_{0i})^{2}-(S_{ii})^{2},\qquad
c^{(2)}_{i}=4(T_{0i})^{2}-(T_{ii})^{2},\qquad i=1,2.$

As previously we will be interested in the degenerated Lax matrices of the
following form

$\displaystyle L(u)=L^{(1,2)}(u)C,$

where the matrix $C=\operatorname{diag}(c_{11},c_{22})$. The generating
function of integrals of motion is

$\displaystyle I(u)=\operatorname{tr}L(u).$

More explicitly, we will have that

$\displaystyle I(u)=\frac{1}{u^{2}-1}\big{(}I_{0}u^{2}+I_{1}u+I_{2}\big{)},$

where the Hamiltonians $I_{k}$ are written as follows

$\displaystyle
I_{0}=(c_{11}(2S_{01}-S_{11})(2T_{01}-T_{11})+c_{22}(2S_{02}-S_{22})(2T_{02}-T_{22})),$
$\displaystyle
I_{1}=(c_{11}\bigl{(}-(2S_{01}-S_{11})(2T_{01}+T_{11})+(2T_{01}-T_{11})(2S_{01}+S_{11})+4T_{12}S_{21}\bigr{)}$
$\displaystyle\hphantom{I_{1}=}{}+c_{22}\bigl{(}-(2S_{02}-S_{22})(2T_{02}+T_{22})+(2T_{02}-T_{22})(2S_{02}+S_{22})-4T_{21}S_{12}\bigr{)}),$
$\displaystyle
I_{2}=-(c_{11}(2S_{01}+S_{11})(2T_{01}+T_{11})+c_{22}(2S_{02}+S_{22})(2T_{02}+T_{22})).$

On the four-dimensional level surface of the Casimir functions the integrals
$I_{1}$ and $I_{2}$ are functionally independent.

####  5.4.2 Separated variables

Let, as previously, the functions $A(u)$ and $B(u)$ be defined as follows

$\displaystyle A(u)=L_{11}(u),\qquad B(u)=L_{21}(u).$

For the considered $N=2$-case of the Lax matrix we obtain

$\displaystyle
A(u)=\frac{c_{11}}{u^{2}-1}\bigl{(}(2S_{01}-S_{11})(2T_{01}-T_{11})u^{2}+\bigl{(}(2T_{01}-T_{11})(2S_{01}+S_{11})$
$\displaystyle\hphantom{A(u)=}{}-(2S_{01}-S_{11})(2T_{01}+T_{11})+4T_{12}S_{21}\bigr{)}u+(2S_{01}+S_{11})(2T_{01}+T_{11})\bigr{)},$
$\displaystyle
B(u)=\frac{((-2T_{01}+T_{11})S_{12}+(2S_{02}-S_{22})T_{12})u-(2T_{01}+T_{11})S_{12}-(2S_{02}+S_{22})T_{12}}{2\big{(}u^{2}-1\big{)}}.$

We define the (quasi)canonical coordinates in our non-standard manner

$\displaystyle A(x_{i})=0,\qquad p_{i}=B(x_{i}),$

i.e., $x_{1}$, $x_{2}$ are the solutions of the following quadratic equation

$\displaystyle(2S_{01}-S_{11})(2T_{01}-T_{11})x^{2}+\bigl{(}(2T_{01}-T_{11})(2S_{01}+S_{11})$
$\displaystyle\qquad{}-(2S_{01}-S_{11})(2T_{01}+T_{11})+4T_{12}S_{21}\bigr{)}x+(2S_{01}+S_{11})(2T_{01}+T_{11})=0,$

and

$\displaystyle
p_{i}=\frac{((-2T_{01}+T_{11})S_{12}+(2S_{02}-S_{22})T_{12})x_{i}-(2T_{01}+T_{11})S_{12}-(2S_{02}+S_{22})T_{12}}{2\big{(}x_{i}^{2}-1\big{)}},$

$i=1,2$.

By the virtue of our general theory the coordinates $x_{i}$, $p_{i}$ are
quasi-canonical

$\displaystyle\\{x_{i},x_{j}\\}=0,\qquad\\{p_{i},p_{j}\\}=0,\qquad\\{x_{i},p_{i}\\}=x_{i}p_{i},\qquad\forall\,i,j=1,2.$
(5.12)

and (in the case $c_{22}=0$) satisfy the following equations of separation

$\displaystyle\big{(}I_{0}x_{i}^{2}+I_{1}x_{i}+I_{2}\big{)}=0,\qquad i=1,2,$
(5.13)

The coordinates of separation $x_{i}$ became the action coordinates in this
case.

####  5.4.3 The reconstruction formulae

In order to completely resolve the problem in the case $N=2$ it is necessary
to be able to reconstruct the dynamical variables via the separated
coordinates and Casimir functions, i.e., it is necessary to resolve the
following system of twelve linear-quadratic equations

$\displaystyle(2S_{01}-S_{11})(2T_{01}-T_{11})x_{i}^{2}+\bigl{(}(2T_{01}-T_{11})(2S_{01}+S_{11})$
$\displaystyle\qquad{}-(2S_{01}-S_{11})(2T_{01}+T_{11})+4T_{12}S_{21}\bigr{)}x_{i}+(2S_{01}+S_{11})(2T_{01}+T_{11})=0,$
(5.14a) $\displaystyle
p_{i}=\frac{((-2T_{01}+T_{11})S_{12}+(2S_{02}-S_{22})T_{12})x_{i}-(2T_{01}+T_{11})S_{12}-(2S_{02}+S_{22})T_{12}}{2\big{(}x_{i}^{2}-1\big{)}},$
$\displaystyle\qquad i=1,2,$ (5.14b) $\displaystyle
C_{1}=2(S_{11}S_{22}-2S_{12}S_{21}-4S_{01}S_{02}),$ (5.14c) $\displaystyle
C_{2}^{2}=(2S_{01}+S_{11})(2S_{02}+S_{22}),$ (5.14d) $\displaystyle
K_{1}=2(T_{11}T_{22}-2T_{12}T_{21}-4T_{01}T_{02}),$ (5.14e) $\displaystyle
K_{2}^{2}=(2T_{01}+T_{11})(2T_{02}+T_{22}).$ (5.14f) $\displaystyle
c_{1}=4(S_{01})^{2}-(S_{11})^{2},$ (5.14g) $\displaystyle
c_{2}=4(S_{02})^{2}-(S_{22})^{2},$ (5.14h) $\displaystyle
k_{1}=4(T_{01})^{2}-(T_{11})^{2},$ (5.14i) $\displaystyle
k_{2}=4(T_{02})^{2}-(T_{22})^{2},$ (5.14j)

on the twelve variables $S_{0i}$, $S_{ij}$, $T_{0i}$ $T_{ij}$, $i,j=1,2$. Here
$C_{i}$, $K_{i}$, $c_{i}$, $k_{i}$, $i=1,2$ are some fixed values of the
Casimir functions.

For the purpose of simplicity we will hereafter consider the restriction of
our quadratic Poisson algebra to the trigonometric Sklyanin algebra case,
i.e., we will put

$\displaystyle S_{01}=S_{02},\qquad S_{22}=-S_{11},\qquad T_{02}=T_{01},\qquad
T_{22}=-T_{11}.$

After such the reduction one can neglect the last four equations in the system
(5.14) due to the fact that the Casimirs $c_{i}$, $k_{i}$ become dependent on
the Casimirs $C_{i}$, $K_{i}$.

The following theorem holds true.

######  Theorem 5.7.

  1. $(i)$

The mechanical coordinates $S_{0i}$, $T_{0i}$, $S_{ij}$, $T_{ij}$ are
parametrized by the separated coordinates $x_{i}$, $p_{i}$ and the values of
the Casimir functions $C_{i}$, $K_{i}$, $i=1,2$ as follows

$\displaystyle
S_{01}=-\frac{C_{2}}{4(D_{1}D_{2})^{\frac{1}{2}}}\bigl{(}(x_{1}-1)(x_{1}+1)^{2}\big{(}4C_{1}x_{2}+C_{2}^{2}+C_{2}^{2}x_{2}^{2}\big{)}p_{1}$
$\displaystyle\hphantom{S_{01}=}{}-(x_{2}-1)(x_{2}+1)^{2}\big{(}C_{2}^{2}x_{1}^{2}+C_{2}^{2}+4C_{1}x_{1}\big{)}p_{2}\bigr{)},$
(5.15a) $\displaystyle
S_{11}=-\frac{C_{2}}{2(D_{1}D_{2})^{\frac{1}{2}}}\big{(}(x_{1}+1)(x_{1}-1)^{2}\big{(}4C_{1}x_{2}+C_{2}^{2}+C_{2}^{2}x_{2}^{2}\big{)}p_{1}$
$\displaystyle\hphantom{S_{11}=}{}-(x_{2}+1)\big{(}x_{2}-1)^{2}(C_{2}^{2}x_{1}^{2}+C_{2}^{2}+4C_{1}x_{1}\big{)}p_{2}\big{)},$
(5.15b) $\displaystyle
S_{21}=-\frac{iK_{2}c_{11}}{8(x_{1}x_{2})^{\frac{1}{2}}(D_{1}D_{2})^{\frac{1}{2}}}(x_{1}-x_{2})\big{(}4C_{1}x_{2}+C_{2}^{2}+C_{2}^{2}x_{2}^{2}\big{)}$
$\displaystyle\hphantom{S_{21}=}{}\times\big{(}C_{2}^{2}x_{1}^{2}+C_{2}^{2}+4C_{1}x_{1}\big{)},$
(5.15c) $\displaystyle
S_{12}=-\frac{2i(x_{1}x_{2})^{\frac{1}{2}}}{(x_{1}-x_{2})c_{11}K_{2}(D_{1}D_{2})^{\frac{1}{2}}}\bigl{(}(x_{1}^{2}-1)^{2}\big{(}4C_{1}x_{2}+C_{2}^{2}+C_{2}^{2}x_{2}^{2}\big{)}p_{1}^{2}$
$\displaystyle\hphantom{S_{12}=}{}-2\big{(}x^{2}_{2}-1\big{)}(x^{2}_{1}-1)\big{(}C_{2}^{2}x_{1}x_{2}+2C_{1}x_{1}+2C_{1}x_{2}+C_{2}^{2}\big{)}p_{2}p_{1}$
$\displaystyle\hphantom{S_{12}=}{}+\big{(}x^{2}_{2}-1\big{)}^{2}\big{(}C_{2}^{2}x_{1}^{2}+C_{2}^{2}+4C_{1}x_{1}\big{)}p_{2}^{2}\bigr{)},$
(5.15d) $\displaystyle
T_{01}=\frac{i(x_{2}-1)(x_{1}-1)K_{2}}{4(x_{1}x_{2})^{\frac{1}{2}}(D_{1}D_{2})^{\frac{1}{2}}}\bigl{(}(x_{1}+1)x_{1}\big{(}4C_{1}x_{2}+C_{2}^{2}+C_{2}^{2}x_{2}^{2}\big{)}p_{1}$
$\displaystyle\hphantom{T_{01}=}{}-(x_{2}+1)x_{2}\big{(}C_{2}^{2}x_{1}^{2}+C_{2}^{2}+4C_{1}x_{1}\big{)}p_{2}\big{)},$
(5.15e) $\displaystyle
T_{11}=\frac{i(x_{1}+1)(x_{2}+1)K_{2}}{2(x_{1}x_{2})^{\frac{1}{2}}(D_{1}D_{2})^{\frac{1}{2}}}\bigl{(}x_{1}(x_{1}-1)\big{(}4C_{1}x_{2}+C_{2}^{2}+C_{2}^{2}x_{2}^{2}\big{)}p_{1}$
$\displaystyle\hphantom{T_{11}=}{}-x_{2}(x_{2}-1)\big{(}C_{2}^{2}x_{1}^{2}+C_{2}^{2}+4C_{1}x_{1}\big{)}p_{2}\bigr{)},$
(5.15f) $\displaystyle
T_{21}=-\frac{c_{11}}{8(x_{1}-x_{2})\big{(}x^{2}_{1}-1\big{)}\big{(}x^{2}_{2}-1\big{)}p_{2}p_{1}C_{2}x_{1}x_{2}(D_{1}D_{2})^{\frac{1}{2}}}$
$\displaystyle\hphantom{T_{21}=}{}\times\bigl{(}x_{1}^{2}\big{(}x^{2}_{1}-1\big{)}^{2}\big{(}4C_{1}x_{2}+C_{2}^{2}+C_{2}^{2}x_{2}^{2}\big{)}^{2}\big{(}{-}K_{2}^{2}x_{2}^{2}+4K_{1}x_{2}-K_{2}^{2}\big{)}p_{1}^{2}$
$\displaystyle\hphantom{T_{21}=}{}-2x_{1}x_{2}\big{(}x^{2}_{2}-1\big{)}\big{(}x^{2}_{1}-1\big{)}\big{(}4C_{1}x_{2}+C_{2}^{2}+C_{2}^{2}x_{2}^{2}\big{)}\big{(}C_{2}^{2}x_{1}^{2}+C_{2}^{2}+4C_{1}x_{1}\big{)}$
$\displaystyle\hphantom{T_{21}=}{}\times\big{(}{-}x_{1}K_{2}^{2}x_{2}+2x_{1}K_{1}+2K_{1}x_{2}-K_{2}^{2}\big{)}p_{2}p_{1}$
$\displaystyle\hphantom{T_{21}=}{}+x_{2}^{2}\big{(}x^{2}_{2}-1\big{)}^{2}\big{(}C_{2}^{2}x_{1}^{2}+C_{2}^{2}+4C_{1}x_{1}\big{)}^{2}\big{(}{-}K_{2}^{2}x_{1}^{2}+4x_{1}K_{1}-K_{2}^{2}\big{)}p_{2}^{2}\bigr{)},$
(5.15g) $\displaystyle
T_{12}=\frac{2(x_{1}-x_{2})\big{(}x^{2}_{1}-1\big{)}\big{(}x^{2}_{2}-1\big{)}p_{2}p_{1}C_{2}}{c_{11}(D_{1}D_{2})^{\frac{1}{2}}},$
(5.15h) $\displaystyle S_{01}=S_{02},\qquad S_{22}=-S_{11},\qquad
T_{02}=T_{01},\qquad T_{22}=-T_{11},$ (5.15i)

where

$\displaystyle
D_{1}=\big{(}x^{2}_{1}-1\big{)}\big{(}4C_{1}x_{2}+C_{2}^{2}+C_{2}^{2}x_{2}^{2}\big{)}p_{1}-\big{(}x^{2}_{2}-1\big{)}\big{(}C_{2}^{2}x_{1}^{2}+C_{2}^{2}+4C_{1}x_{1}\big{)}p_{2},$
$\displaystyle
D_{2}=x_{1}\big{(}x^{2}_{1}-1\big{)}\big{(}4C_{1}x_{2}+C_{2}^{2}+C_{2}^{2}x_{2}^{2}\big{)}p_{1}-x_{2}\big{(}x^{2}_{2}-1\big{)}\big{(}C_{2}^{2}x_{1}^{2}+C_{2}^{2}+4C_{1}x_{1}\big{)}p_{2}.$

  2. $(ii)$

If the Poisson relations among $p_{i}$ and $x_{j}$ have quasi-canonical form
(5.12) then the variables $S_{0i}$, $T_{0i}$, $S_{ij}$, $T_{ij}$ given by
(5.15) satisfy the Poisson brackets (5.11).

###### Idea of the proof.

Item (i) of the theorem is proven by the direct calculations. The item (ii)
follows from our general theory and can be also checked by the direct
verifications. ∎

####  5.4.4 The Abel-type equations

In the case $c_{22}=0$, $c_{11}\neq 0$ the Abel equations written for the
separated coordinates $x_{i}$ are trivial due to the fact that $x_{i}$ in this
case are expressed via the integrals of motion. The non-trivial Abel-type
equations are written for the conjugated momenta $p_{i}$

$\displaystyle\dfrac{1}{2x_{1}I_{0}+I_{1}}\frac{1}{p_{1}}\frac{{\rm
d}p_{1}}{{\rm d}t_{1}}+\dfrac{1}{2x_{2}I_{0}+I_{1}}\frac{1}{p_{2}}\frac{{\rm
d}p_{2}}{{\rm d}t_{1}}=1,$
$\displaystyle\dfrac{1+x_{1}^{2}\dfrac{c_{11}^{2}c_{1}c_{2}}{I^{2}_{2}}}{2x_{1}I_{0}+I_{1}}\frac{1}{x_{1}p_{1}}\frac{{\rm
d}p_{1}}{{\rm
d}t_{1}}+\dfrac{1+x_{2}^{2}\dfrac{c_{11}^{2}c_{1}c_{2}}{I^{2}_{2}}}{2x_{2}I_{0}+I_{1}}\frac{1}{x_{2}p_{2}}\frac{{\rm
d}p_{2}}{{\rm d}t_{1}}=0,$
$\displaystyle\dfrac{1}{2x_{1}I_{0}+I_{1}}\frac{1}{p_{1}}\frac{{\rm
d}p_{1}}{{\rm d}t_{2}}+\dfrac{1}{2x_{2}I_{0}+I_{1}}\frac{1}{p_{2}}\frac{{\rm
d}p_{2}}{{\rm d}t_{2}}=0,$
$\displaystyle\dfrac{1+x_{1}^{2}\dfrac{c_{11}^{2}c_{1}c_{2}}{I^{2}_{2}}}{2x_{1}I_{0}+I_{1}}\frac{1}{x_{1}p_{1}}\frac{{\rm
d}p_{1}}{{\rm
d}t_{2}}+\dfrac{1+x_{2}^{2}\dfrac{c_{11}^{2}c_{1}c_{2}}{I^{2}_{2}}}{2x_{2}I_{0}+I_{1}}\frac{1}{x_{2}p_{2}}\frac{{\rm
d}p_{2}}{{\rm d}t_{2}}=1.$

Introducing the angle variables $p_{i}=\exp{\phi_{i}},i=1,2$ we obtain that

$\displaystyle\phi_{1}=\phi_{1}^{0}-\frac{ic_{11}K_{2}C_{2}(x_{1}-x_{2})}{2(x_{1}x_{2})^{\frac{1}{2}}}\big{(}t_{1}-t^{0}_{1}\big{)}+\frac{1}{2}ic_{11}K_{2}C_{2}(x_{1}x_{2})^{\frac{1}{2}}\big{(}t_{2}-t^{0}_{2}\big{)},$
$\displaystyle\phi_{2}=\phi_{2}^{0}-\frac{ic_{11}K_{2}C_{2}(x_{2}-x_{1})}{2(x_{1}x_{2})^{\frac{1}{2}}}\big{(}t_{1}-t^{0}_{1}\big{)}+\frac{1}{2}ic_{11}K_{2}C_{2}(x_{1}x_{2})^{\frac{1}{2}}\big{(}t_{2}-t^{0}_{2}\big{)},$

where $x_{i}$, $i=1,2$ are the constants of motion calculated from the
equations (5.13) and we have taken into account that there exists the
following relations among the integrals $I_{0}$, $I_{1}$, $I_{2}$, Casimir
functions $C_{2}$, $K_{2}$ and the action coordinates $x_{1}$, $x_{2}$

$\displaystyle
I_{1}=\frac{ic_{11}(x_{1}+x_{2})K_{2}C_{2}}{(x_{1}x_{2})^{\frac{1}{2}}},\qquad
I_{0}=-\frac{ic_{11}K_{2}C_{2}}{(x_{1}x_{2})^{\frac{1}{2}}},\qquad
I_{2}=-ic_{11}K_{2}C_{2}(x_{1}x_{2})^{\frac{1}{2}}.$

##  6 Conclusion and discussion

In the present paper we have constructed non-standard separated variables for
the classical integrable XXX and XXZ spin chains with the degenerated twist
matrix reversing the role of the functions $A(u)$ and $B(u)$ in the
corresponding separating algebra. We have also obtained the Abel-type
equations for the corresponding coordinates/momenta of separation. We have
shown that for special cases of the degenerated twist matrices our separated
variables can be identified with the action-angle coordinates from the
Liouville theorem.

It would be interesting and physically important to find quantum and higher
rank analogues of the proposed separated variables. These problems are open.

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