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

. 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.


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, . . . , N are a set of (quasi)canonical coordinates such that the following system of equations is satisfied [24] Φ i (x i , p i , I 1 , . . . , I N , C 1 , . . . , C r ) = 0, i = 1, . . . , N, where Φ 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. 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 gl(2)-valued Lax matrices L(u) = 2 i,j=1 L ij (u)X ij , where X ij , i, j = 1, 2 is a standard basis of gl(2) (X ij ) αβ = δ iα δ j,β , have been introduced in [ The constructed variables x i , p i should be (quasi)canonical in order for the theory to work 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 This situation occurs when the Lax matrix satisfy quadratic tensor brackets [7,19] {L(u) ⊗ 1, 1 ⊗ L(v)} = [r(u, v), L(u) ⊗ L(v)]. (1.5) Here r(u, v) = 2 i,j,k,l=1 r ij,kl (u, v)X ij ⊗ 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 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. 2 In the present paper we are going to answer two general questions: 1. For what gl(2) ⊗ 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. 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 There are at least two gl(2) ⊗ 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 c ij X ij is a two by two twist matrix satisfying the following condition and L (ν i ) (u) is the Lax matrix with a simple pole in the point u = ν i , corresponding to i-th cite of the classical spin chain, where the classical spins satisfy the Poisson brackets of gl(2) ⊕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) = tr L(u).
The following theorem holds true: Let the coordinates x i , p i , i = 1, . . . , 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 = 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) = tr L(u) if and only if For the rational XXX case the matrix C = 2 i,j=1 c ij X ij is an arbitrary constant degenerated matrix. In this case the case the equations of separation are written as follows and we distinguish to cases: c 12 = 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 where dx i dt k = {I k , x i }, k = 1, . . . , 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 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 and produce linear differential equations for the corresponding angle coordinates φ i = ln p i .
In the trigonometric XXZ case the constant twist matrix C is diagonal, 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. To find separated variables means to find (at least locally) a set of coordinates x i , p j , i, j = 1, . . . , N such that there exist N relations Φ i (x i , p i , I 1 , . . . , I N , C 1 , . . . , C r ) = 0, i = 1, . . . , N, where C i , i = 1, . . . , r are Casimir functions and the coordinates x i , p j , i, j = 1, . . . , N are canonical In the present paper it will be convenient for us to work with quasi-canonical coordinates satisfying the following Poisson brackets Clearly the variables x i , φ 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 and similar Abel-type equations are satisfied by the momenta of separation These equations are the last step before the integration of the classical equations of motion.

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, . . . , N be zeros of the function B(u) and p i , i = 1, . . . , 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, . . . , 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 Then the Poisson bracket between the functions x i and p j , ∀ i, j = 1, . . . , N are the following If, moreover also the condition holds, then the corresponding Poisson brackets are quasi-canonical, i.e., Remark 2.2. Observe that the coefficients a(u, v), b(u, v), α(u, v), β(u, v), γ(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), α(u, v), β(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

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

The separating functions
Let X ij , i, j = 1, 2 be a standard basis in gl (2) with the commutation relations The gl(2)-valued Lax matrix is written as follows 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:

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] 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. 3 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 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.
Proof . The proposition is proven by direct calculation.
There are at least two gl(2) ⊗ 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 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

Poisson brackets and Lax matrix
Let us now consider the simplest possible case of the standard rational r-matrix 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 where N is arbitrary and the basic one-spin matrices L (i) (u) are written as follows Here ν i = ν j , when i = j, i, j = 1, . . . , N and the Poisson brackets among the coordinates S Hereafter we will be interested in the Lax matrices of the following form where C is an arbitrary constant matrix C =

Integrals and Casimir functions
Let us now consider the integrable system on gl(2) ⊕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) The generating function of integrals of motion is More explicitly, we have that where the Hamiltonians I k are non-homogeneous polynomials of the degree up to k in the dynamical variables. In particular, we have that 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 gl(2) ⊕N algebra by Pauli matrices) using formulae of the following type [24] H n = d n ln I(u) du n 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.

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 where the functions A(u) and B(u) are standardly defined using the Lax matrix L(u) For the Lax operator defined by the formulae (4.2), (4.3) we obtain 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 = 0 one can take A(u) for the role of separating polynomial: it produces N separated coordinates x i .
The following theorem holds true. The equations of separation have in this case the following form 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 . 5 In this case we have But in this way the integrals of motion are also changed -instead of the generating function one obtains a new generating function 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 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 Now it is left to show that they satisfy the equations of separation. We will prove them for any Lax matrix of the form L(u) =L(u)C, whereL(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 matrixL(u) and the matrix C we obtain From this it immediately follows that the following identity holds true

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 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.

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. But, taking into account that c 22 = 0 we will obtain from (4.5) The corollary is proven.

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 where dp i dt k = {I k , x i }, k = 1, . . . , 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, . . . , 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 φ i by we obtain that the equations (4.7) are easily integrated as follows The last equations have the following explicit solution 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 actionangle variables in the corresponding XXX spin chain.

The model
Let us consider the simplest non-trivial example of N = 2. We will have L (1,2) (u) = L (ν 1 ) (u)L (ν 2 ) (u), and the basic one-spin matrices L (i) (u) are written as follows Hereafter for the purpose of convenience we will locate the poles as follows: ν 1 = 1, ν 2 = −1, and set S {T ij , S kl } = 0. (4.8c) As previously we will be interested in the Lax matrices of the following form: where the matrix C is an arbitrary constant degenerated matrix C = 2 i,j=1 c ij X ij , det(C) = 0.
The generating function of integrals of motion is More explicitly, we will have that where the Hamiltonians I k are non-homogeneous polynomials in the dynamical variables c ij (S ij + T ij ), c ii .

Separated variables
Let, as previously, the functions A(u) and B(u) are defined as follows For the considered N = 2 case of the Lax matrix we obtain We will again define the (quasi)canonical coordinates as follows that is x 1 , x 2 are the solutions of the following quadratic equation By virtue of our general theory the coordinates x i , p i are quasi-canonical and satisfy the following equation of separation In the case c 12 = 0, c 22 = 0 these equations of separation degenerate to the form i.e., the coordinates of separation x i became the action coordinates in this case.

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  (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 (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

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 = 0 the Abel equations are written for the separated coordinates x i as follows They are easily resolved in terms of elementary functions.

Abel-type equations: special degenerated case
In case c 12 = 0, c 22 = 0, c 11 = 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 where we have used the following relations among x i and I i Introducing the angle variables p i = exp φ i , i = 1, 2 we obtain that where x i , i = 1, 2 are the constants of motion calculated from the equations (4.10).
5 Classical XXZ spin model

The r-matrix, the basic Lax matrix and quadratic brackets
Let us consider the following r-matrix 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 ν = 0, ν = ∞. Then the Lax matrix L (ν) (u) defined by the formula where the Poisson brackets among the coordinate functions S 0i , S ii , S ij are the following 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 C(u) = det(L(u)).
More explicitly, we will have the following expansion, yielding three Casimir functions C k Besides, there also exist two additional quadratic Casimir functions 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 On its zero level surface one can put and arrive exactly to the trigonometric degeneration of the famous elliptic Sklyanin algebra on the space sl(2) extended with the help of one-dimensional center S 0 = S 01 = S 02 .

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 L (1,2,...,N ) (u) = L (ν 1 ) (u) · · · L (ν N ) (u), (5.4) where the elementary Lax matrices L ν k (u) are given by the formula (5.2) The Poisson brackets among the functions S 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 L(u) = L (1,2,...,N ) (u)C, (5.6) where C is a constant matrix, in a standard way I(u) = tr(L(u)).

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 where the functions A(u) and B(u) are standardly defined using the Lax matrix L(u) For the Lax operator defined by the formulae (5.4), (5.6) we also obtain where A k , B k are the functions of the dynamical variables S ij and the constants c ij . In particular, we have that It is important that on a generic symplectic leaf one has A 0 = 0, A N = 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. 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 By the virtue of this algebra we obtain in the trigonometric case the quasi-canonical Poisson brackets of the following form 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 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.

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 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 we obtain that the equations (5.9) are easily integrated as follows The explicit solution of (5.10) is constructed in the same manner as in the rational case.

The model
Let us consider the simplest non-trivial example of N = 2. We will have and the basic one-spin matrices L (ν k ) (u) are written as follows Hereafter for the purpose of convenience we will put ν 1 = 1, ν 2 = −1, The Poisson brackets are those of the direct sum of two quadratic algebras (5.3) The Casimir functions of the quadratic Sklyanin-type parenthesis of the direct sum are Besides, there exist also the following quadratic Casimir functions i = 4(T 0i ) 2 − (T ii ) 2 , i = 1, 2.
As previously we will be interested in the degenerated Lax matrices of the following form L(u) = L (1,2) (u)C, where the matrix C = diag(c 11 , c 22 ). The generating function of integrals of motion is I(u) = tr L(u).
More explicitly, we will have that where the Hamiltonians I k are written as follows On the four-dimensional level surface of the Casimir functions the integrals I 1 and I 2 are functionally independent.

Separated variables
Let, as previously, the functions A(u) and B(u) be defined as follows The coordinates of separation x i became the action coordinates in this case.

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 (2S 01 − S 11 )(2T 01 − T 11 )x 2 i + (2T 01 − T 11 )(2S 01 + S 11 ) − (2S 01 − S 11 )(2T 01 + T 11 ) + 4T 12 S 21 x i + (2S 01 + S 11 )(2T 01 + T 11 ) = 0, (5.14a) 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 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.