Affine and Finite Lie Algebras and Integrable Toda Field Equations on Discrete Space-Time

Difference-difference systems are suggested corresponding to the Cartan matrices of any simple or affine Lie algebra. In the cases of the algebras $A_N$, $B_N$, $C_N$, $G_2$, $D_3$, $A_1^{(1)}$, $A_2^{(2)}$, $D^{(2)}_N$ these systems are proved to be integrable. For the systems corresponding to the algebras $A_2$, $A_1^{(1)}$, $A_2^{(2)}$ generalized symmetries are found. For the systems $A_2$, $B_2$, $C_2$, $G_2$, $D_3$ complete sets of independent integrals are found. The Lax representation for the difference-difference systems corresponding to $A_N$, $B_N$, $C_N$, $A^{(1)}_1$, $D^{(2)}_N$ are presented.

(1) 2 , D (2) N these systems are proved to be integrable. For the systems corresponding to the algebras A 2 , A called generalized two-dimensional Toda lattices have very important applications in Liouville and conformal field theories, they are studied in details (see [3,5,6,9,10,11,13,27,28,29,30,34,36] and the references therein). Here the matrix A = {a i,j } is the Cartan matrix of an arbitrary finite or affine Lie algebra. It is known that in the former case system (1.1) is Darboux integrable while in the latter case -S-integrable. The widely known Drinfel'd-Sokolov formalism allows one to construct the Lax representation for the system (1.1) in terms of the Lie algebra canonically associated with the corresponding Cartan matrix A. The problem of finding discrete versions of the system is intensively studied (see, for instance, [2,14,15,16,17,19,20,21,22,24,25,38,39,40]). Recently, in [18] integrable differentialdifference analog of system (1.1) was suggested and r j x = ∂ ∂x r j (n, x). A particular case of (1.2), corresponding to the algebra C N is also studied in [37].
In the present article we study the problem of further discretization of system (1.1), i.e. the problem of finding a rule allowing to assign to any Cartan matrix a system of integrable difference-difference equations approximating in the continuum limit system (1.2) and therefore system (1.1). As it was pointed out in [40] the problem of discretization is important from physical viewpoint, they might have applications in discrete field theory and in quantum physics (see also [22,25,38]). They can also be regarded as difference schemes in numerical computations (see [23]).
Following questions were addressed in [40]: 1) whether there exists an integrable discrete version for any two dimensional Toda field equation (1.1); 2) which kind of algebraic structure (like Lie algebra or Lie group) is naturally related to discrete versions.
The subject has intensively been studied during the last 10-15 years. Various discrete versions of Toda field equations were investigated in literature (see [15,22,24,25,39] and references therein). As an alternative answer to the first question we suggest a fully discrete version of system (1.1) in the following form which evidently approximates (1.1) and (1.2). Here u j = u j (n, m), j = 1, 2, . . . , N , is a set of the field variables. The subindex indicates shifts of the arguments n, m as follows u j i,k := u j (n + i, m + k). The system corresponding to the algebra A N coincides with that found years ego by Hirota (see [21]). Obviously, system (1.3) is invariant under the replacement n ↔ −m. This property is inherited from the fact that system (1.1) is invariant under the change x ↔ y.
The main result of the present article is in formulating of the conjecture below and proving it for numerous examples of Lie algebras.

Conjecture.
a) If A is the Cartan matrix of a semi-simple Lie algebra then (1.3) is Darboux integrable, in other words, it admits a complete set of integrals in both directions. Roughly speaking Darboux integrable difference-difference equations can be reduced to ordinary difference equations.
b) If A is the Cartan matrix of an affine Lie algebra then (1.3) is S-integrable, i.e. it can be integrated by means of the inverse scattering transform method. It is generally accepted that existence of generalized symmetries or Lax pairs indicates S-integrability.
Part a) of the conjecture is proved for the cases: A 2 , B 2 , C 2 , G 2 , D 3 by finding complete sets of integrals. Systems corresponding to the algebras A N , B N , C N are studied by using Lax pair in Section 6. In the cases A 2 , B 2 (C 2 ), G 2 the exponential system (1.3) reads as follows Integrals in both directions for system (1.4) are given in Section 2.
Since the map defined in (1.3) converts any N ×N matrix to a system of difference-difference equations one can easily specify the form of systems corresponding to any Cartan matrix canonically related to a finite or affine Lie algebra both of classical series or exceptional. For instance, for the algebra D 3 with the Cartan matrix Complete sets of integrals for this system is given in Section 2.
For the Kac-Moody algebra A For the difference-difference systems corresponding to the algebras A 2 , A 1 , A 2 generalized symmetries are found, for A 2 the symmetries have usual form, for A (1) 1 , A 2 they have nonlocal form (or, they are of hyperbolic type, see Definition 1.5 below). In the literature generalized symmetries are regarded as a criterion of integrability. Thus one concludes that systems (1.5), (3.1) provide new examples of 2 × 2 S-integrable quad graph models.
Stress that the number of the field variables in all three systems (1.1), (1.2), (1.3) coincides with the rank of the corresponding Lie algebra.
It is known that any system of difference-difference equations of hyperbolic type admits a pair of characteristic Lie algebras which are effectively evaluated (see [14]). Characteristic Lie algebras of system (1.3) are closely connected with the Lie algebra canonically related to the given Cartan matrix A. In our opinion this relation could allow one to answer the second question in [40] (see the list of questions above).
The problem of developing discrete versions of the Drinfel'd-Sokolov formalism is also challenging since today systems of discrete equations are very popular. They have a large variety of application in theoretical physics, in discrete geometry, in the theory of tau functions in lattice Toda field equations. The present article would also provide an "experimental" background for creating a discrete theory parallel to the Drinfel'd-Sokolov formalism.
The article is organized as follows. In Section 1.1 notions of the integrals and generalized symmetries of both evolutionary and hyperbolic type for quad graph equations are defined. Sufficient condition of complete set of integrals is proved. In Section 2 a method of discretization of Darboux integrable models preserving integrability is discussed. The method is explained in details by an example of the Liouville equation [19]. Then it is applied to the system (1.2) to get its difference-difference analog. The case B 2 is studied in more details. These simulations allowed us to guess the formula (1.3). Complete sets of independent integrals for the systems Generalized symmetries are evaluated for the systems corresponding A 2 , A 2 in Section 3. Special attention is paid to hyperbolic type symmetries which provide a semi-discrete version of the consistency around a cube property of quad graph equations [1]. In Section 4 the concept of the characteristic Lie algebra (see [14]) of discrete models is briefly discussed. Here a complete description of such algebra is obtained for the system of difference-difference equations A 2 .
In Section 5 cutting off conditions for the Hirota equation preserving integrability are studied by using the method suggested in [15]. It turned out that some of the systems in the class (1.3) can be obtained from the Hirota equation by imposing proper boundary conditions. Since the cutting off conditions are compatible with the Lax representation one can derive together with the reduced discrete system also its Lax pair. In Section 5 the Lax pairs for the difference- N are found. In Section 6 an algorithm is suggested to look for integrals via the Lax pair. In Section 7 widely known periodical reduction of the Hirota equation is discussed. The Lax pair for this system found in [41] is rewritten in terms of the Cartan-Weyl basis for the algebra A (1) N . Emphasize that the difference-difference system obtained as a periodical reduction and that given by the formula (1.3) corresponding to A

Integrals and symmetries for quad graph systems
Consider a system of quad graph equations of general form where u = u n,m is a vector-function depending on two integers and ranging on C N : u = (u 1 , u 2 , . . . , u N ) T . As usually we request that equation (1.6) can be solved with respect to any of the arguments u n,m , u n+1,m , u n,m+1 , u n+1,m+1 . In other words there exists a set of functions H (±1,±1) such that u n+1,m+1 = H 1,1 (u n,m , u n+1,m , u n,m+1 ), Define the following standard set of dynamical variables which consists of the variable u n,m and its shifts u n+k,m and u n,m+l where k, l ∈ Z S = {u n+k,m , u n,m+l : k, l ∈ Z}.
By [u] we denote a finite set of dynamical variables, for instance notation h = h([u]) means that function h depends on a finite number of dynamical variables. Define shift operators D m , D n acting due to the rules ii) Integrals of the form F = F (n) (I = I(m)) are called trivial.
iii) Equation admitting N non-trivial independent integrals in each direction is called Darboux integrable.
iv) Set of integrals is called independent if none of them can be expressed through the other integrals and their shifts.
It can be easily proved that m-integrals do not depend on the variables u n,m+l with l = 0 and similarly n-integrals do not depend on u n+k,m where k = 0. Now formulate a very simple and convenient sufficient condition of complete set of integrals.
Theorem 1.2. Let us given a set of m-integrals of the form Suppose that for any j and for all n, m at least one of the derivatives ∂u i differs from zero and the condition det holds for all n, m. Then the integrals constitute a complete set of integrals.
Proof . Suppose in contrary that the set of integrals (1.7) is not independent. Then at least one of the integrals, say for definiteness I (1) is a function of the form depending on the other integrals and their shifts. Differentiating (1.9) with respect to u i we obtain the following equalities The latter can be regarded as a linear algebraic system with the coefficient matrix { ∂u i } and a solution 1, ∂Q ∂I (2) , . . . , ∂Q which obviously is not trivial. Therefore according to the wellknown theorem the determinant of the coefficient matrix should vanish. But it contradicts (1.8). Thus our assumption that the set of integrals is dependent is not correct. Theorem is proved. Remark 1.3. Condition (1.8) actually means that jacobian of the map u → I = I (1) , . . . , I (N ) differs from zero for any value of the variables u, D n u, D 2 n u, . . . , D γ n u ranging in a domain. Here γ = max{γ j }. Note that (1.8) is not a necessary condition for independent integrals. In the Example 6.1 we have a pair of independent integrals I (1) and I (2) for which condition (1.8) is violated. is called generalized symmetry of (1.6) if the following compatibility condition is satisfied It can be shown that function G([u]) is a solution of linearized equation There are systems of the form (1.6) which do not admit evolutionary type symmetries, however they admit symmetries of more complicated structure. We call them hyperbolic type symmetries. Such type symmetries for (1.6) consist of two semi-discrete equations Hyperbolic type symmetries are constructed in Section 3. They are nothing else but non-local symmetries (see, for instance, [35]) rewritten in a more convenient form. In the other hand side existence of hyperbolic type symmetries (1.10) for equation (1.6) can be regarded as a semidiscrete generalization of the well-known property of consistency around a cube [1,4,32] which is approved to be a criterion of integrability.
2 Systematic approach to the problem of discretization of the Darboux integrable systems The problem of finding integrable discretizations of the integrable partial differential equation is very complicated and not enough studied. The same is true for evaluating the continuum limit for discrete models. In [19] an effective algorithm of discretization (as well as evaluation of the continuum limit) of Darboux integrable equations is suggested based on the integrals. In this section we discuss the essence of the algorithm and apply it to exponential type systems. Note that sets of integrals for the systems corresponding to A 2 , B 2 , G 2 , D 3 considered in this section as easily proved by applying sufficient condition given in Theorem 1.2 are independent.

Explanation of the method with example of Liouville equation
Its well-known that the famous Liuoville equation u xy = e u admits integrals in both directions: Indeed, it is very easy to check that D y W = 0 and D xW = 0.

Application of the algorithm of discretization
to the system corresponding to the algebra B 2 Apply the reasonings above to the system (1.1) corresponding to the algebra B 2 . The first step has already been done in [18], where the differential-difference system was found admitting two independent n-integrals having the form which are also integrals of fully continuous system (1.1) corresponding to the same algebra. It was shown in [18] that system (2.8) admits also x-integrals The goal of this section is to construct a difference-difference system having the same functions F (1) and F (2) as their m-integrals.
To make the formulas shorter we change the variables where a, b are new unknowns. The given m-integrals in these variables read Formulas (2.9) define integrals for a semi-discrete system (2.8) while (2.11) defines integrals for a certain difference-difference system, that is why the variables in (2.11) should be labeled by a double index, however we omitted here and below in this section the second index, because its value is zero for all considered variables. Now we will look for the equations desired in the variables a, b in the form (2) and bring them to the following form

Substitute the integrals (2.11) into the equations for m-integrals D
By differentiating equation (2.12b) with respect to the variables a 3 , b 3 we get equations Similarly, differentiation of (2.12a) with respect to a 3 , b 3 yields From these equations we get By differentiating (2.12) with respect to a 2 we obtain The last two equations can be rewritten as follows By getting rid of the variable ∂ a 1 D n g 1 in these equations we find The last implication is due to the assumption that F (1) and F (2) are integrals of the lowest order. As a result we get equations After some rescaling we can put C = C 1 = 1. Finally, we end up with the system of discrete equations corresponding to the algebra B 2 . Turn back to the original variables u 1 , u 2 (see (2.10)) By construction the last system admits pair of integrals (2.9) in which the second index for the variables u 1 , u 2 is omitted, since its value is the same for all variables. To find the integrals in the other direction one uses the discrete symmetry n ↔ −m of the system.

System corresponding to the algebra A 2
The Cartan matrix of A 2 is The discrete system for A 2 looks as follows For system (2.13) m-integrals are To find the integrals in the other direction one uses the discrete symmetry n ↔ −m of the system.

System corresponding to the algebra G 2
The Cartan matrix of G 2 is The discrete system for G 2 is of the form Its m-integrals are To find the integrals in the other direction one uses the discrete symmetry n ↔ −m of the system.

System corresponding to the algebra D 3
The Cartan matrix of D 3 is The discrete system for D 3 looks as Its m-integrals are By applying Theorem 1.2 one can prove that these integrals provide a complete set of independent integrals. To find the integrals in the other direction one uses the discrete symmetry n ↔ −m of the system.

Symmetries of discrete systems
In this section we demonstrate that the discrete systems admit generalized symmetries. For the case of the simple Lie algebras the systems admit local symmetries while for the case of affine algebras the symmetries are nonlocal.

Generalized symmetries for the system A 2
Higher symmetries for system corresponding to A 2 are found by using the method suggested in [12] (see also [26]). The lowest order symmetry is of the form .
The next order symmetry depends on arbitrary functions f (x, y), g(x) of two and respectively one variable and has the form

Evaluation of hyperbolic type symmetries for the system
The method for searching nonlocal symmetries is illustrated with the following example corresponding to the algebra A 1 with the Cartan matrix Recall that the continuous version of the system is It can be proved that system (3.2) does not have any local generalized symmetry, however it has nonlocal generalized symmetries, the simplest one can be represented in terms of s = u + v, r = u − v as follows Here F is an arbitrary function and W = s xx − r 2 x is the y-integral of system (3.2). Note that years ago in [9] it was observed that equation r t = r xxx − 2r 3 x is consistent with the system (3.2). For the fully discrete analogue (3.1) of the system (3.2) we have a very similar situation. The system (3.1) does not have any local generalized symmetry. One has to look for a nonlocal symmetry. The main trouble arising in this case is connected with the guessing of the structure of non-locality and the form of the symmetry.
Rewrite system (3.1) in the form We will look for the hyperbolic type symmetry (see Definition 1.5) of (3.3) in such a form Here f 1 , g 1 , f , g are unknown functions. Right hand side of (3.4) corresponds to the function G in (1.10). The function G is evaluated in terms of G below by means of the compatibility conditions. From (3.4) we can find After differentiation (3.1) with respect to t by means of the systems (3.4), (3.5) we get Here the right hand sides do not depend on the derivatives of dynamical variables with respect to t. By applying the operator D n to both sides of (3.6) we obtain System of equations (3.6), (3.7) is a system of linear algebraic equations with unknowns ∂ t a 0,0 , If the determinant of this system is different from zero, then due to the Cramer's rule the system has unique solution, and therefore the searched generalized symmetry is local. This output is in contradiction with our previous study proving the absence of local symmetries. Thus the determinant should be zero Vanishing of the determinant implies In virtue of (3.8) equations (3.6), (3.7) take the form We can see that the left hand sides of the first and third equations as well as the left hand sides of the second and fourth equations of the last system are identical. Comparison of their right hand sides gives a system of two equation for f , g Differentiate (3.10) and (3.11) with respect to a 3,0 , b 3,0 , b −1,0 or a −1,0 and get We use Continuing this way one can find dependence of the functions f , g on the variables a −1,0 , a 2,0 , a 0,0 , a 1,0 , b 0,0 , b 1,0 ), a 0,0 , a 1,0 , b 0,0 , b 1,0 ).
For the functions f 1 , g 1 from the same system (3.10), (3.11) we obtain the following nonhomogeneous equations Differentiating (3.15) with respect to the variable a 2,0 we get From this equation we can find Differentiating (3.15) with respect to b 2,0 we get  0 , a 1,0 , b 1,0 ). Substitute in (3.14) the expression found instead of f 1 and differentiate with respect to a 2,0 to get a 0,0 , b 1,0 ).
For the functions f 3 , g 3 we have a system We see that (3.16) is satisfied identically if one chooses Thus we find the final form of the symmetry searched Here C 1 , C 2 , C 3 are arbitrary constants. From (3.9) one can find the function G, then (1.10) looks like  By analogy with the continuous case we have a differential constraint completely consistent with system (3.3), which is obtained from the last system by applying the operator (D n − 1) −1 to the difference of the two equations Under the Cole-Hopf type transformation 3) converts to the following onê This transformation brings our hyperbolic type symmetry to generalized symmetry of usual form for the new system (3.19)

Hyperbolic type symmetries for the system A
(2) 2 The Cartan matrix of A 2 is The discrete system for A Here C 1 , C 2 , C 3 are arbitrary constants. Second part of hyperbolic type symmetry (function G) can be evaluated automatically from the compatibility conditions. And for combination we have a local constraint Algebra L m is called characteristic m-algebra. Obviously, operators X 1 , X 2 are the first-order linear differential operators, or vector fields. The operators Y 1 , Y 2 can also be rewritten as vector fields of the form We use the following lemma to show that m-algebra is of finite dimension.
Lemma 4.1. Suppose that the vector field One can easily check that Taking commutators of the vector fieldsX 1 ,X 2 ,Ỹ 1 ,Ỹ 2 we get vector fields Lemma 4.2.
Lemma can be proved by direct calculations. It allows to derive the following table of commutators which shows that the characteristic Lie algebra L m is of dimension seven: Here the following notations are used Due to the reasonings above any m-integral F (. . . , a 0,0 , b 0,0 , a 1,0 , b 1,0 , a 2,0 , b 2,0 , . . . ) should satisfy equations Solving system (4.3), it is enough to assume that F depends on b 0,0 , a 0,0 , b 1,0 , a 1,0 , b 2,0 , a 2,0 or, alternatively, F depends on a −1,0 , b 0,0 , a 0,0 , b 1,0 , a 1,0 , b 2,0 . Under such assumptions system (4.3) generates two systems of the first-order linear partial differential equations which can be solved by Jacobi method. By solving these systems we find two independent m-integrals

Cutting of f constraints for the Hirota equation and discrete Zakharov-Shabat systems
In this section we construct Lax pairs for discrete systems corresponding to Cartan matrices of N and A 1 . To this end we impose cutting off constrains for the Hirota equation compatible with its Lax pair.
It is well known that Hirota chain admits the Lax pair consisting of two linear discrete equations [7] ψ j Here the lower indices mean as previously shifts of the arguments, and the upper index enumerates the field variables t j and eigenfunctions ψ j . Exclude from the system of equations (5.2) all the eigenfunctions with the upper index different from j. As a result one gets a linear discrete hyperbolic equation for ψ j It is remarkable that by construction formulas ( and equation (5.3) transforms to an equation

then Lax pair (5.4) transforms to a Lax pair
and equation (5.5) transforms to an equation To this end we evaluate the Laplace invariants of these equations.
Recall that the Laplace invariants of a discrete hyperbolic type equation of the form are given by (see [2,8,31,33]) By virtue of these formulas the invariants K 1ψ , K 2ψ and K 1g , K 2g of equations (5.3), (5.7) are, respectively, It is known that two linear hyperbolic type equations are related to one another by a linear transformation only if their corresponding Laplace invariants are equal. Evidently in generic case coincidence of the Laplace invariants generates two constraints on the field variables t j = t j (n, m). Only for some special cases it gives only one constraint. We are interested in such special cases. For instance pair of equations K 1ψ (n, m, j) = K 1g (n + 1, m, j − 1), K 2ψ (n, m, j) = K 2g (n + 1, m, j − 1) is equivalent to the constraint which is interpreted as a cutting off boundary condition for the chain (5.1). For simplicity we put j = 0, so the boundary condition becomes t −1 1,0 = t 1 0,1 . Following [15] we can construct a Lax pair for the reduced chain. Under the boundary condition (5.8) we have coincidence of the invariants K 1ψ (n, m, j) = K 1g (n + 1, m, j − 1), K 2ψ (n, m, j) = K 2g (n + 1, m, j − 1), and we can relate the eigenfunctions We study the finite reductions of the chain (5.1) on a finite interval N L ≤ j ≤ N R . The reduction is obtained by imposing the boundary conditions at the left end-point j = N L t N L −1 1,0 = t N L +1 0,1 (5.9) and respectively at the right end-point j = N R (5.10) First we concentrate on the left end-point. Due to the reasonings above the eigenfunctions should satisfy the following gluing conditions These conditions allow one to close the Lax equations at the left end-point From (5.2), (5.6) we have

R. Garifullin, I. Habibullin and M. Yangubaeva
To derive similar equations at the point N R we use the right end-point constraint (5.10), for which we have These conditions allow one to close the Lax equations at the right end-point From (5.2), (5.6) we have The Lax pair found above is not convenient to work with because the operators contain shifts in opposite directions. Below we show that it can be rewritten in a usual form.
Shift of equation (5.11) forward with respect to the variable n brings it to the form Shift of equation (5.12) forward with respect to the variable m brings it to the form So we have the following system of linear equations 14) In the next subsection we gather these equations to a matrix form.

Lax pair for systems corresponding to the algebras D
(2) Imposing of non-degenerate boundary conditions (5.9), (5.10) at both end-points N L = 0 and N R = N leads to the system corresponding to the matrix D Write the set of equations (5.13)-(5.18) in the form of a Lax pair for system (5.19). Denote the eigenvector as follow P = (ψ 0 , ψ 1 , . . . , ψ N −1 , g 0 , g 1 , . . . ,

R. Garifullin, I. Habibullin and M. Yangubaeva
It is straightforward to check that the compatibility condition of the equations is equivalent the system (5.19).
Example 5.2. Consider a particular case when N L = 0 and N R = 1 The system corresponds to algebra A 1 . Its Lax pair is given by (5.20) with the matrices Example 5.3. If one imposes condition (5.9) for N L = 0 and (5.10) for N R = 2, then The Lax pair is of the form (5.20) where A and B are 4 × 4 matrices Remark 5.4. In Example 5.2 we give a Lax pair realized in 2 × 2 matrices, while general formula (5.20) generates 3 × 3 matrices. The matter is that in the Lax pair obtained directly from (5.20) we made in this case some additional reduction.

Lax pair for systems corresponding to the algebras A N
Instead of boundary condition (5.8) we can use also the degenerate boundary conditions of the form at the left end-point and t N R +1 = 0, t N R = 1 (5.22) at the right end-point. The degenerate boundary conditions imply that the corresponding eigenfunctions are zero: g N L −1 = 0 and ψ N R = 0.
In order to obtain the system corresponding to the Cartan matrix A N −1 we cut off the Hirota chain by imposing degenerate boundary conditions (5.21) at the point N L = 0 and (5.22) at N R = N . The resulting reduction is as follows In this case our algorithm gives the Lax pair found years ago in [21]. In order to formulate it introduce the eigenvector P = (ψ 0 , ψ 1 , . . . , ψ N −1 ) T and N × N matrices It is easy to check that the compatibility condition of the equations leads to the system (5.23).

Lax pair for systems corresponding to the algebras B N
We impose boundary condition (5.21) for N L = 0 and (5.10) for N R = N . The resulting reduction is as follows System (5.24) can be rewritten in form of (1.3) by changing the variables t j = e −u j . It corresponds to the algebra B N . The system admits a Lax pair. Let us denote P = (ψ 0 , ψ 1 , . . . , ψ N −1 , g 0 , g 1 , . . . , g N −1 ) T and introduce 2N × 2N matrices Then according to our general scheme the compatibility condition of the equations P 1,0 = AP, P 0,1 = BP leads to the system (5.24).
Remark 5.5. The system B N can be obtained from the system A 2N −1 by imposing the cutting off constraint of the form (see Lemma 5.1 above).
6 Method of f inding integrals from Lax representation for the systems corresponding to the Cartan matrices A N , B N In this section we show that the Lax pair allows one to generate integrals for the systems corresponding to the simple Lie algebras A N , B N . Concentrate on m-integrals. Due to the definition we have two different expression for the shifted eigenvector P k,1 , k ≥ 1 P k,1 = V k,0 U k−1,0 U k−2,0 · · · U 1,0 U 0,0 P, and similarly Comparison of these two formulas yields Due to the triangular structure of the matrices V k,0 and V 0,0 the map converting any upper triangular matrix X to a matrixX = V k,0 XV −1 0,0 leaves unchanged the element of the matrix X located at the right upper corner: (X) 1,N =X 1,N . Thus the corresponding element, denote it through I (k−N ) (k > N ), of the upper triangular matrix U k−1,0 U k−2,0 · · · U 1,0 U 0,0 is an mintegral. In such a way we get a set of integrals I (1) , I (2) , . . . , I (N ) . Examples below show that they constitute a complete set of m-integrals, however we are not able to prove this fact in general. In a similar way one can find integrals in the other direction.

Recall its Lax pair
where P = (ψ 0 , ψ 1 , ψ 2 ) T and Evaluate the elements at the right upper corner for the following two products U 2,0 U 1,0 U 0,0 and U 3,0 U 2,0 U 1,0 U 0,0 and find two independent m-integrals where the second index for the variables t 1 , t 2 is omitted. Since the integral I (2) is too complicated one can choose a more simple onẽ By using Theorem 1.2 one can check that integrals I (1) and I (2) provide a complete set of integrals.
In a similar way integrals for the system (5.24), corresponding to the algebra B N , are constructed. Let us consider a matrix Φ of dimension 2N × 2N where E 11 is the unity matrix of dimension N , E 12 , E 21 are matrices of dimension N with zero entries, E 22 is a matrix of dimension N with the unity adverse diagonal. By an automorphism X → ΦXΦ −1 the matrices (5.25) and (5.26) are transformed to the triangular matrices R. Garifullin, I. Habibullin and M. Yangubaevā By an automorphism acting as follows X → F −1 XF matrix (6.2) is transformed to a lower triangular matrix for which all diagonal entries are equal to the unity. Here F is a 2N × 2N matrix of the form  Here the second index for the variables t 1 , t 2 is omitted, since its values is zero for all considered variables. It can be proved by using Theorem 1.2 that integrals I (1) and I (2) constitute a complete set of integrals.

Periodic boundary conditions
In this section we discuss briefly the well known periodically closed reduction of the Hirota chain (see for more details and the references [41]). Close the chain (5.1) by imposing the periodical boundary conditions Close the Lax pair (5.2) by setting the conditions on the eigenfunctions As a result we get a finite system of the form t 0 0,0 t 0 1,1 − t 0 1,0 t 0 0,1 = t N 1,0 t 1 0,1 , t j 0,0 t j 1,1 − t j 1,0 t j 0,1 = t j−1 1,0 t j+1 0,1 , which is closely connected with the Cartan matrix A N . Boundary conditions (7.1) reduce the sequence of linear discrete equations (5.2) to a Lax pair for the reduced system (7.2). Introduce the eigenvector P = (ψ 0 , ψ 1 , . . . , ψ N ) T and N +1×N +1 matrices If the functions t 0 , t 1 , . . . , t N solve the system (7.2) then the overdetermined system of linear equations P 1,0 = U P, P 0,1 = V P (7.3) is compatible (see [41]).