Generalized fermionic discrete Toda hierarchy

Bi-Hamiltonian structure and Lax pair formulation with the spectral parameter of the generalized fermionic Toda lattice hierarchy as well as its bosonic and fermionic symmetries for different (including periodic) boundary conditions are described. Its two reductions --N=4 and N=2 supersymmetric Toda lattice hierarchies-- in different (including canonical) bases are investigated. Its r-matrix description, monodromy matrix, and spectral curves are discussed.

The present paper continues studies of the above-mentioned hierarchies and is addressed to yet unsolved problems of constructing their periodic counterparts, bi-Hamiltonian structure in different (including canonical) bases, (2m × 2m)-matrix and 4 × 4-matrix (3 × 3-matrix) Lax pair descriptions with the spectral parameter, r-matrix approach, and spectral curves.
The structure of this paper is as follows. In section 2.1, starting with the zero-curvature representation we introduce the 2D generalized fermionic Toda lattice equations and describe their two reductions related to the N = (2|2) and N = (0|2) supersymmetric Toda lattice equations. Then, in section 2.2, we construct the bi-Hamiltonian structure of the 1D generalized fermionic Toda lattice hierarchy, and its fermionic and bosonic Hamiltonians.
Sections 3 and 4 are devoted to the 1D N = 4 and N = 2 supersymmetric Toda lattice hierarchies, respectively. We construct their bi-Hamiltonian structure in sections 3.1 and 4.1, fermionic symmetries in section 3.2, and in sections 3.3 and 4.2, we investigate a transition to the canonical basis which spoils a number of supersymmetries.
In section 5, we consider periodic supersymmetric Toda lattice hierarchies. Thus, in section 5.1, we construct the (2m × 2m)-matrix zero-curvature representation with the spectral parameter for the periodic 2D generalized fermionic Toda lattice hierarchy. Then, in section 5.2, we obtain the bi-Hamiltonian structure of its one-dimensional reduction. In section 5.3, we construct the (4 × 4)-matrix Lax pair representation of this hierarchy, calculate its r-matrix, and analyze monodromy matrix. We next calculate its spectral curves in section 5.4. In section 5.5, we give a short summary of the (3 × 3)-matrix Lax pair representation and the r-matrix formalism for the periodic 1D N = 2 Toda lattice hierarchy, and calculate spectral curves of the latter. In section 5.6, we discuss periodic Toda lattice equations in the canonical basis and their fermionic symmetries.
Our starting point is the following zero-curvature representation: for the infinite matrices Here, z 1 and z 2 are the bosonic coordinates (∂ 1,2 ≡ ∂ ∂z 1,2 ); the matrix entries d j , c j (ρ j , γ j ) are the bosonic (fermionic) fields with Grassmann parity 0 (1) and length dimensions [d j ] = −2, [c j ] = −1, [ρ j ] = −3/2 and [γ j ] = −1/2. The zero-curvature representation (2.1) leads to the following system of evolution equations with respect to the bosonic evolution derivatives ∂ 1,2 : Keeping in mind that in the bosonic limit (i.e., when all fermionic fields are put equal to zero) these equations describe a system of two decoupled bosonic 2D Toda lattices, we call equations (2.3) the 2D generalized fermionic Toda lattice equations. Our next goal is to describe fermionic symmetries of the 2D generalized fermionic Toda lattice equations (2.3). Before doing so let us first supply the fields (d j , c j , γ j , ρ j ) with boundary conditions. In what follows we consider the boundary conditions of the following four types: IV ). d j = d j+n , c j = c j+n , γ j = γ j+n , ρ j = ρ j+n , n ∈ Z. (2.4) The first three types specify the behavior of the fields at the lattice points at infinity while the boundary condition of the fourth type is periodic and corresponds to the closed 2D generalized fermionic Toda lattice. For the boundary conditions I) and II) (2.4) the above described equations (2.3) possess the N = (2|2) supersymmetry. Indeed, in this case there exist four fermionic symmetries of equations (2.3) D 1 1 d j = g j−1 ρ j + g j ρ j−1 , D 1 2 d j = (−1) j (g j−1 ρ j − g j ρ j−1 ), D 1 1 c j = g j γ j−1 + g j+1 γ j , D 1 2 c j = (−1) j (g j+1 γ j − g j γ j−1 ), D 1 1 ρ j = −∂ 1 g j , D 1 2 ρ j = (−1) j ∂ 1 g j , D 1 1 γ j = g j − g j+2 , D 1 2 γ j = (−1) j (g j+2 − g j ) (2.5) where D 1 1 , D 1 2 , D 2 3 and D 2 4 are the fermionic evolution derivatives; g j denotes the infinite product with the properties g j g j−1 = d j and . Now using eqs. (2.3) and (2.5)-(2.6) one can easily check that the bosonic and fermionic evolution derivatives satisfy the algebra of the N = (2|2) supersymmetry which can be realized via where z a (a = 1, 2) and θ s , θ p (s = 1, 2; p = 3, 4) are the bosonic and fermionic evolution times of the N = (2|2) superspace, respectively. Looking at equations (2.5)-(2.6) one can see that they are not consistent with the boundary conditions III) (2.4). Thus, it is impossible to simultaneously satisfy the boundary conditions for the fields g j entering into eqs. (2.5) and eliminate the fields c j from eq. (2.3) in order to get the conventional form of the 2D N = (2|2) supersymmetric Toda lattice equations [12] together with their fermionic N = (2|2) symmetries (2.13) In order to derive the second representation, let us introduce a new notation for the fields at odd and even values of the lattice coordinate j (2.14) and rewrite eqs. (2.3), (2.5-2.6) in the following form: where e j ,ē j are the composite fields which obey the equations The reductionb of eqs. (2.15) leads to the 2D N = (0|2) supersymmetric Toda lattice equations [12,5]. One can easily see that fermionic symmetries (2.16) are not consistent with this reduction, while fermionic symmetries (2.17) are consistent and form the algebra of the N = (0|2) supersymmetry.

Bi-Hamiltonian structure of the 1D generalized fermionic Toda lattice hierarchy
Our further purpose is to construct a bi-Hamiltonian structure of the generalized fermionic Toda lattice equations (2.3) (and, consequently, originating from them eqs. (2.12) and (2.15)) in one-dimensional space when all the fields depend on only one bosonic coordinate z = z 1 + z 2 . This task was solved in [6] for the 1D N = 2 Toda lattice hierarchy obtained by reduction (2.20) of the 1D generalized fermionic Toda lattice hierarchy. Here we solve this task for the original 1D generalized fermionic Toda lattice hierarchy.
At the reduction to one-dimensional space, the zero-curvature representation (2.1) can identically be rewritten in the form of the Lax-pair representation Using the Lax pair representation (2.22), it is easy to derive the general expression for bosonic Hamiltonians which are in involution via the standard formula The first two of them have the following explicit form: A bi-Hamiltonian system of evolution equations can be represented in the following general form: where t H k are the evolution times, q j denotes any field from the set q i = {d i , c i , ρ i , γ i } and the brackets {, } 1(2) are appropriate Poisson brackets corresponding to the first (second) Hamiltonian structure. Using eqs. (2.25) and the 2D generalized fermionic Toda lattice equations (2.3) at the reduction to one-dimensional space (2.21) -the 1D generalized fermionic Toda lattice equations as well as Hamiltonians (2.24), we have found the first two Hamiltonian structures of the hierarchy. As the result, we have the following explicit expressions: for the first and Thus, one concludes that the corresponding recursion operator is hereditary like the operator obtained from the compatible pair of Hamiltonian structures.
We have checked that the one-dimensional reduction (2.21) of the fermionic symmetries (2.5)-(2.6) and the equations for the composite fields g i (2.7) can also be represented in a bi-Hamiltonian form with fermionic Hamiltonians S s,k and Hamiltonian structures (2.27) and (2.28) where D t S s,k are the fermionic evolution derivatives. In section 3.2 we show how fermionic Hamiltonians can be derived in an algorithmic way, but now let us only mention that there are four infinite towers of fermionic Hamiltonians S s,k (s = 1, 2, 3, 4; k ∈ N) and present without any comments only explicit expressions for the first few of them For completeness we also present the nonzero Poisson brackets of the composite field g i (2.7) with other fields of the hierarchy which are useful when producing fermionic Hamiltonian flows Now we have all necessary ingredients to derive Hamiltonian flows of the 1D generalized Toda lattice hierarchy. Let us end this section with a few remarks.
First, the Hamiltonians H 1 (2.24) and S s,1 (2.32) give trivial flows via the first Hamiltonian structure (2.27) because they belong to the center of the algebra (2.27) Second, while the densities corresponding to the fermionic Hamiltonians S p,k (2.32) have a nonlocal character with respect to the lattice indices, the fermionic flows (2.29) have no nonlocal terms.
Finally, the algebras of the first and second Hamiltonian structures (2.27)-(2.28) together with eqs. (2.33) possess a discrete inner automorphism f which transforms nontrivially only fermionic fields and the first and the second Hamiltonian structures, where only nonzero brackets are presented. For the first nontrivial bosonic and fermionic flows one obtains in a standard way, using eqs. (2.25), (2.31) and (3.1)-(3.3), with the N = 4 supersymmetry transformations (3.8) and it transforms the flows (3.4)-(3.5) and Hamiltonians (3.1) as follows:

Fermionic Hamiltonians
The above-described 1D N = 4 supersymmetric Toda lattice hierarchy is a bi-Hamiltonian system, and it includes both bosonic and fermionic flows which are generated via bosonic and fermionic Hamiltonians. The bosonic Hamiltonians are produced by means of formula (2.23), while the origin of the fermionic Hamiltonians is rather mysterious so far. In this section, we deduce general expressions generating fermionic Hamiltonians. The N = (2|2) Toda lattice equations (2.12) can be derived as a subsystem of more general N = (1|1) 2D supersymmetric Toda lattice (TL) hierarchy defined via the following Lax pair representation [13]: All details concerning the N = (1|1) 2DTL hierarchy can be found in [5,13], here we only explain the notation. The generalized graded bracket operation [..., ...}, entering into eqs.
where O * (m) denotes the m-fold action of the involution * on the operator O. Equations (3.12) are written for the composite Lax operators k,j ≡ v k,j ) are the functionals of the original bosonic u 2k,j ,v 2k,j and fermionic u 2k+1,j ,v 2k+1,j lattice fields which parameterize the Lax operators L ± (3.12). The operator e l∂ (l ∈ Z) acts on these fields as the discrete lattice shift k,j can be obtained through the representation of the composite Lax operators (L ± ) m * (3.12), (3.14) in terms of the Lax operators L ± (3.12). The fields u k,j , v k,j depend on the bosonic t ± 2n and fermionic t ± 2n+1 times, and D ± 2n (D ± 2n+1 ) in eq. (3.12) means bosonic (fermionic) evolution derivatives with the algebra which can be realized via (3.17) Now using the above-described definitions one can derive flows for the functionals u corresponding to the Lax pair representation (3.12). Thus, we obtain [5,13] where in the right-hand side of these equations, all the fields {u k,j } with k < 0 must be set equal to zero.
The N = (2|2) supersymmetric 2DTL equation belongs to the system of equations (3.18)-(3.25). In order to see that, let us consider eqs. (3.21 Then, eliminating the field u 1,j from eqs. (3.26)-(3.27) we obtain where g j (γ ± j ) are the bosonic (fermionic) fields and | means the t ± 1 → 0 limit, eq. (3.28) coincides with (2.12) 2k+1,j ) as a sum of all their diagonal elements of the trivial shift operator with l = 0 (e 0∂ = 1) multiplied by the factor (3.31) One can easily verify that the main property of supertraces is indeed satisfied for the case of the generalized graded bracket operation (3.13). Using this definition of the supertrace and Lax pair representation (3.12) one can easily obtain conserved Hamiltonians of the N = (1|1) 2DTL hierarchy From this formula it is obvious that all bosonic Hamiltonians corresponding to even values of m are trivial H α 2n = 0 like a supertrace of the generalized graded bracket operation, while fermionic Hamiltonians at odd values of m are not equal to zero H α 2n−1 = 0 in general. Using eqs. (3.14) we obtain more explicit superfield formulae for the latter which in terms of superfield components look like The functionals u is the fermionic integral in eqs. (3.1). A more complicated problem is to obtain the next fermionic integral s + 2 (u). First, using eq. (3.12) one can find the explicit form of the functional u respectively. From eq. (3.38) it follows that the consequence of eqs. (3.39) and (3.27) is and, at last, from eq. (3.27) one can find that The two remaining series of fermionic Hamiltonians in eqs. (3.1) can easily be derived from the obtained ones if one applies the automorphism transformations (3.10)

Transition to the canonical basis for the N=4 Toda lattice equations
Our next task is to rewrite the N=4 Toda lattice equations (3.4) in a canonical basis where these equations admit a Lagrangian formulation that is important in connection with the quantization problem. Let us introduce the new basis {x j , p j , where i is the imaginary unity and we suppose that the new fields go to zero at infinity In terms of the new coordinates the first Hamiltonian structure (3.2) becomes canonical and the Hamiltonians (3.1) take the following form: The Hamiltonian H 2 generates the following equations via the first Hamiltonian structure (3.47) Following the standard procedure one can derive the Lagrangian L and the action S The variation of the action S with respect to the fields {x j , χ − j , χ + j } produces the equations of motion (3.49) for them with reversed sign of time (∂ → − ∂ ∂t ) where the momenta p j are replaced by (−1) j ∂ ∂t x j . One important remark is in order. There is no one-to-one correspondence between the phase space bases {g j , c j , γ + j , γ − j } and {x j , p j , χ + j , χ − j } (3.45). Transformation (3.45) can rather be treated as a reduction of the primary phase space {g j , c j , γ + j , γ − j } to the subspace {x j , p j , χ + j , χ − j } with a smaller symmetry. Indeed, the direct consequence of eqs. (3.45)-(3.46) is the following constraints on the original fields Equations (3.51) restrict the phase space of (3.4) and change the symmetry properties of the latter. The first manifestation of such a restriction is the fact that Hamiltonians H 1 and S n,1 (3.48) no longer belong to the center of the first Hamiltonian structure and generate the following nontrivial flows via the algebra (3.47): Furthermore, conditions (3.51) break the N = 4 supersymmetry transformations (3.5) and in order to restore the N = 4 supersymmetry, one needs to impose the following additional constraints on the fields: However, the following N = 2 supersymmery transformations in terms of fermionic flows (3.5): are consistent with the constraints (3.51) and provide the N = 2 sypersymmetry for the infinite Toda lattice in the canonical basis (3.49)

Equation (3.49) can be represented in the superfield form
where Φ j is the bosonic N = 2 superfield with the components Here | means the θ ± → 0 limit and D ± are the fermionic covariant derivatives In order to rewrite the second Hamiltonian structure (3.3) in terms of the new fields {x j , p j , χ + j , χ − j }, we invert (3.45) and find the Poisson brackets between the fields {x j , p j , χ + j , χ − j } using relations (3.3). Equations (3.59) contain three arbitrary parameters c and c ± . However, the second Hamiltonian structure (3.3) is not consistent with constraints (3.51) in general, and it is not guaranteed a priori that the Poisson brackets obtained in such a way obey the Jacobi identities. The test of the Jacobi identities shows that the Poisson brackets obtained form a closed algebra only at c = 1, c ± = 0 and have the following explicit form: where we have introduced the notation The 1D N = 2 Toda lattice hierarchy was proposed and studied in detail in [6]. In this section, we reproduce its bi-Hamiltonian description [6] Substituting this solution into eqs. (2.15) at ∂ 2 = ∂ 1 = ∂ we arrive at the following equations: Fermionic symmetries (2.16) become inconsistent after reductionb j = 0 because in this case the fields e j (2.18) become singular. As concerns fermionic symmetries (2.17), they are consistent and take the following form: (4.4) The system (4.3) is supplied with the boundary conditions I) and III) (2.4). Let us recall that for the boundary conditions III) (2.4) the system (2.15) does not possess any supersymmetry (see the paragraph with eqs. (2.10)). In terms of fields (2.14) the boundary conditions I) and III) (2.4) are respectively. Therefore, we conclude that system (4.3) possesses N = 2 supersymmetry only for the boundary conditions Ib) (4.5), while for the boundary conditions IIb) (4.5) it is not supersymmetric.
The first (2.27) and second (2.28) Hamiltonian structures in the basis {b j ,b j , a j ,ā j , α j ,ᾱ j , β j ,β j } (2.14) look like and respectively. One can easily see that the algebras (4.6) and (4.7) are consistent with the reduction constraints (2.20) and (4.2), so the 1D N = 2 supersymmetric Toda lattice equations (4.3) can be represented as a bi-Hamiltonian system with the first Hamiltonian structure (4.8) and the second Hamiltonian structure where when calculating we have substituted the reduction constraints (2.20) and (4.2) into the original algebras (4.6) and (4.7). Let us also present a few first bosonic and fermionic Hamiltonians of the 1D N = 2 supersymmetric Toda lattice hierarchy obtained from Hamiltonians (2.24) and (2.32) using reduction constraints (2.20) and (4.2) (4.10)

Transition to the canonical basis for the 1D N=2 Toda lattice equations
The transition to the canonical basis for the 1D N = 2 supersymmetric Toda lattice equations (4.3) with non-periodic boundary conditions is possible only for the boundary conditions IIb) (4.5). As we have already mentioned, in this case the system (4.3) is not supersymmetric, nevertheless it can serve as a basement for the building of the periodic N = 2 Toda lattice equations in the canonical basis. In this section, we briefly discuss the representation of the system (4.3) in the canonical basis. Following paper [6], let us introduce the new basis {x j , p j , ξ j ,ξ j , η j ,η j } in the phase space with the zero boundary conditions at infinity lim j→±∞ {x j , p j , ξ j ,ξ j , η j ,η j } = 0. (4.12) In this basis the first Hamiltonian structure becomes canonical while the second Hamiltonian structure takes a more complicated form where c is an arbitrary parameter andc = 1 or 0. One can trace the origin of these parameters if one writes down the most general form of the inverse transformations (4.11) From the Jacobi identities one can fix the parameterc to be 1 or 0, while the second parameter c is left arbitrary.
In the canonical basis (4.11) the bosonic Hamiltonians (4.10) become (4.16) and they generate, via the first (4.13) and second (4.14) Hamiltonian structures, the following equations [6]: The parameters c,c do not affect equations (4.17) via the second Hamiltonian structure (4.14). Let us also present the Lagrangian L and the action S One can easily verify that the variation of the action S with respect to the fields {x j , ξ j ,ξ j , η j ,η j } produces equations of motion (4.17) for them with reversed sign of time (∂ → − ∂ ∂t ) where the momenta p j are replaced by − ∂ ∂t x j .

Periodic 2D generalized fermionic Toda lattice equations
The n-periodic 2D generalized fermionic Toda lattice equations (2.3) are characterized by the boundary conditions IV ) (2.4). This system has completely different symmetry properties for odd and even values of the period n. From now on we concentrate on the case with even value n = 2m of the period. The 2m-periodic 2D generalized fermionic Toda lattice equations (2.3) admit the zerocurvature representation where w is the spectral parameter of length dimension [w] = −m. Now, following paper [15] let us give some definitions concerning supermatrices. For any n × n supermatrix F one can define the Grassmann parity of rows and columns as p row (i) ≡ p(F i,1 ) and p col (j) ≡ p(F 1,j ), respectively, where p(F i,j ) is the Grassmann parity of the matrix element F i,j . For the matrices L ± 2m one has p row (i) = p col (i) ≡ p(i). Matrix F has certain Grassmann parity, if the expression does not depend on i and j. For even n = 2m the matrices L ± n have Grassmann parity p(L ± 2m ) = 0, while for odd n = 2m + 1 matrices L ± 2m+1 have no definite parity and, therefore, for odd n the zero-curvature representation (5.1) (as well as the Lax pair representation in one-dimensional space) does not make sense.

Bi-Hamiltonian structure of the periodic 1D generalized fermionic Toda lattice hierarchy
The periodic 1D generalized fermionic Toda lattice equations (2.26) with the 2m-periodic boundary conditions IV ) (2.4) can be reproduced via the following Lax pair representation: where L ± 2m are defined according to eqs. (5.2).
The 2m-periodic 1D generalized fermionic Toda lattice equations (2.26) possess the bi-Hamiltonian structure which can easily be derived from the first and second Hamiltonian structures (2.27) and (2.28), if one makes changes there according to the substitution and change the sum limits in the Hamiltonians (2.24) as Thus, the first and second Hamiltonians structures explicitly are and respectively. Bosonic integrals of motion of the 2m-periodic 1D generalized Toda lattice hierarchy can be derived via the following general formula: Here H 2m k are the bosonic Hamiltonians (5.6), and I 2m k and I 2m k are the additional conserved quantities. We analyzed attentively the quantities I 2m k for the case m = 2, 3 and found that I 2m k can be decomposed into a sum of a few terms which are conserved separately and besides H 2m k contain two more independent integrals of motion of length dimension k < −1 where U 2m k and V 2m k are additional bosonic integrals of motion and S 2m 3 and S 2m 4 are fermionic integrals (see eqs. (5.12)). Our conjecture is that formulae (5.10)-(5.11) are valid not only for the values m = 2, 3 for which they were actually calculated, but also for an arbitrary value of m.
The first fermionic Hamiltonians (2.32) in the 2m-periodic case become Note that in the periodic case the fields g j are connected with the fields d j via the irreversible relation d j = g j g j−1 . For the fields g j there are equations (2.30) and it seems reasonable to consider (2.26), (2.29) and (2.30) as a single joined system of equations. In this case, the system possesses the N = 4 supersymmetry and has additional bosonic integrals of motion V 2m k and U 2m k which can be derived using automorphism (2.35) We suppose that the Hamiltonians (5.12) are the only independent fermionic integrals of motion which exist for the 2m-periodic 1D generalized fermionic Toda lattice equations (2.26). Thus, we have checked that higher fermionic Hamiltonians of length dimensions -3/2 and -5/2 in the 2m-periodic case for m = 2 become composite and can be expressed via the fermionic Hamiltonians (5.12) and bosonic integrals of motion as a sum of composite terms.

The r-matrix formalism
There is another approach to reproduce bosonic integrals of motion [16]. Let us consider the 2m-periodic auxiliary linear problem for the wave functions ψ j such that ψ j+2m = wψ j . One can check that (5.14)-(5.15) are equivalent to the following linear problem: 5.17) and the 1D generalized fermionic Toda lattice equations (2.26) result from the consistency condition ∂L j (λ) = U j+1 (λ)L j (λ) − L j (λ)U j (λ) (5.18) of the linear system (5.16). Let us note that the 4 × 4-matrix Lax operator L j (λ) (5.17) has the fermionic Grassmann parity p(L j (λ)) = 1, according to the definition (5.3). The transformations (2.14) to the new basis {a j ,ā j , b j ,b j , α j ,ᾱ j , β j ,β j } in the space of the functions {c j , d j , ρ j , γ j } together with the new definitions allow us to rewrite eqs. (5.14)-(5.18) in the following equivalent form: Now, we introduce a new basis {p j ,p j , x j ,x j , η j ,η j , ξ j ,ξ j } in the space of the functions {a j ,ā j , b j ,b j , α j ,ᾱ j , β j ,β j }, such that the first Hamiltonian structure (2.27) becomes canonical and after gauge transformation the linear problem in eqs. (5.21) looks like where all matrix entries of the matrix L j (λ) are defined at the same lattice node(!) Here, we note that the (4 × 4)-matrix Lax operator L j (λ) (5.27) has the bosonic Grassmann parity p( L j (λ)) = 0, according to the definition (5.3), and sdet L j (λ) = 1. (5.28) The matrices L j (λ) obey the r-matrix Poisson brackets which are equivalent to the algebra (5.25) and is the permutation matrix. The Grassmann parity function p(j) = 0(1) for bosonic (fermionic) rows and columns of a supermatrix, and for the supermatrix L j (λ) (5.27) we have p(1) = p(2) = 0, p(3) = p(4) = 1. In (5.29) we have used the graded tensor product of two even supermatrices A and B [15] ( Let us note that the operator L j (λ) (5.19) can be represented as a product of two fermionic operators l j (λ) andl j (λ) where we have introduced the supermatrix W j which we define as (5.37) Then, after the gauge transformation (5.26) the Lax operator L j (λ) (5.27) has the form of the product of two fermionic operators l j (λ) and l j (λ) and each of them is defined at the same lattice node(!) It would be interesting to establish r-matrix Poisson bracket relations (if any) between the fermionic supermatrices l j (λ), l j (λ). In order to rewrite the monodromy matrix in terms of the original fields {d j , c j , ρ j , γ j } one can perform the inverse gauge transformations where all the matrix entries are expressed in terms of the fields {d j , c j , ρ j , γ j }. In (5.40) the periodicity property Ω m+1 = Ω 1 of the gauge transformation matrix (5.26) has been used. Relation (5.35) is also true for the monodromy matrix T (λ) because of the relation strT k m (λ) = str T k m (λ). In other words, we have shown that m integrals of the motion being expressed in terms of the original fields {d j , c j , ρ j , γ j } are in involution. However, as the decomposition (5.9) shows, for the 2m-periodic problem there are more than m integrals of motion. In order to obtain them, let us investigate the decomposition The first several coefficients J 2m p have the following explicit form: One can see that the additional integrals V 2m 2 and U 2m 2 are contained in the coefficient J 2m 2 in a different combination than in I 2m 2 (5.10). This is the way to detect them. We suppose that for integrals of higher length dimensions the situation is the same: there are three independent coefficients H 2m k , I 2m k and J 2m k of length dimension k in decompositions (5.9) and (5.41) and each of them is an independent integral of motion.
Having bosonic and fermionic integrals of motion for the 2m-periodic 1D generalized fermionic Toda lattice equations (2.26) it is easy to obtain integrals of motion for the periodic N = 4 (3.4) and N = 2 (4.3) Toda lattice equations. This can be done, respectively, using transformations (2.11) and (2.14) together with the reduction constraints (2.20) and (4.2).

5.5
Reduction: r-matrix approach and spectral curves for the periodic 1D N=2 Toda lattice hierarchy and it is equivalent to the following linear problem: As concerns the periodic 1D N = 2 Toda lattice equations (4.3), they are equivalent to the lattice zero-curvature representation In the canonical basis {x j , p j , α j ,ᾱ j , β j ,β j } (4.11) after the gauge transformation eqs. (5.51) take the form and is defined at the same lattice node. The matrices L j (λ) have the Grassmann parities p(1) = p(2) = 0, p(3) = 1 and obey the same r-matrix Poisson bracket relations (5.29) with the appropriate r matrix. The equation for eigenvalues is defined by the characteristic function where all the coefficients are expressed in terms of the invariants of the monodromy matrix µ k = str T k m = ( T k m ) 11 + ( T k m ) 22 − ( T k m ) 33 , k = 1, 2, 3, From the relation sdet L j (λ) = λ −1 it follows that sdet T m = λ −m and, consequently, σ 3 = −λ m σ 2 .

Periodic Toda lattice equations in the canonical basis
In the previous subsections we considered the 1D generalized fermionic Toda lattice equations  (−1) j ( 1 2 p 2 j − e x j −x j−2 − e x j −x j−1 (χ − j−1 − (−1) j χ + j )(χ − j + (−1) j χ + j−1 )) (5.66) and the canonical first Hamiltonian structure (3.47). Following the standard procedure, one can derive the Lagrangian L and the action S The variation of the action S with respect to the fields {x j , χ − j , χ + j } produces the equations of motion (3.49) for them with reversed sign of time (∂ → − ∂ ∂t ) where the momenta p j are replaced by (−1) j ∂ ∂t x j . The situation with the system (4.3) with the m-periodic boundary conditions is completely different. Let us recall that the infinite system It appears that one can preserve one supersymmetry if one modifies the transition to the canonical basis (4.11) as follows: where s is an arbitrary parameter. In this basis the m-periodic first Hamiltonian structure still has the canonical form (4.13), and using the Hamiltonian it generates the following equations: One can standardly generate the Lagrangian L and the action S The variation of the action S with respect to the fields {x j , ξ j ,ξ j , η j ,η j } produces the equations of motion (5.73) for them with reversed sign of time (∂ → − ∂ ∂t ) where the momenta p j are replaced by − ∂ ∂t x j . If, in addition to the momenta, the fields η j andη j are also eliminated from (5.73) by means of the corresponding equations expressing them in terms of the fields {x j , ξ j ,ξ j } and their derivatives, the remaining equations become ∂ 2 x j = e x j+1 −x j − e x j −x j−1 − ξ j ∂ξ j + ξ j+1 ∂ξ j+1 , ∂(e x j−1 ∂ξ j ) = e x j−1 ξ j−1 − e x j ξ j , ∂(e −x j ∂ξ j ) = e −x jξ j+1 − e −x j−1ξ j . (5.75) It is interesting to remark that the dependence of (5.75) on the parameter s completely disappears. For every m the system (5.73) possesses the N = 1 supersymmetry at the unique value s = ±1/2, and the supersymmetry flows (D 2 ± = ∓∂ t ) are