We describe 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. 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.

By exhibiting the corresponding Lax pair representations we propose a wide class of integrable two-dimensional (2D) fermionic Toda lattice (TL) hierarchies which includes the 2D N = (2|2) and N = (0|2) supersymmetric TL hierarchies as particular cases. Performing their reduction to the one-dimensional case by imposing suitable constraints we derive the corresponding 1D fermionic TL hierarchies. We develop the generalized graded R-matrix formalism using the generalized graded bracket on the space of graded operators with an involution generalizing the graded commutator in superalgebras, which allows one to describe these hierarchies in the framework of the Hamiltonian formalism and construct their first two Hamiltonian structures. The first Hamiltonian structure is obtained for both bosonic and fermionic Lax operators while the second Hamiltonian structure is established for bosonic Lax operators only. We propose the graded modified Yang-Baxter equation in the operator form and demonstrate that for the class of graded antisymmetric R-matrices it is equivalent to the tensor form of the graded classical Yang-Baxter equation.

The zero-curvature representation of the periodic fermionic two-dimensional Toda lattice equations is constructed. It is shown that their reduction to the one-dimensional space is N = 4 supersymmetric and possesses a bi-Hamiltonian structure. Their r-matrix description, monodromy matrix, and spectral curves are discussed.

By exhibiting the corresponding Lax-pair representations, we propose a wide class of integrable twodimensional (2D) fermionic Toda lattice (TL) hierarchies, which includes the 2D N =(2|2) and N =(0|2) supersymmetric TL hierarchies as particular cases. We develop the generalized graded R-matrix formalism using the generalized graded bracket on the space of graded operators with involution generalizing the graded commutator in superalgebras, which allows describing these hierarchies in the framework of the Hamiltonian formalism and constructing their first two Hamiltonian structures. We obtain the first Hamiltonian structure for both bosonic and fermionic Lax operators and the second Hamiltonian structure only for bosonic Lax operators.

In this letter we consider the nonlinear realizations of the classical Polyakov's algebra W(2) 3 . The coset space method and the covariant reduction procedure allow us to deduce the Boussinesq equation with interchanged space and evolution coordinates. By adding one more space coordinate and introducing two copies of the W (2) 3 algebra, the same method yields the sl(3, R) Toda lattice equations. *