Supersymmetric Toda Lattice Hierarchies

The origin of the bosonic and fermionic solutions, constructed in [1, 2, 3], to the symmetry equations corresponding to the two-dimensional bosonic and N = (2 2) supersymmetric Toda lattices is established, and algebras of the corresponding symmetries are derived. The conjecture regarding an N = (2|2) superfield formulation of the N = (2|2) supersymmetric Toda lattice hierarchy, proposed in [16], is proved. The two-dimensional N = (0|2) supersymmetric Toda lattice hierarchy is proposed and its N = (0|2) superfield formulation is discussed. Bosonic and fermionic solutions to the symmetry equation corresponding to the two-dimensional N = (0|2) supersymmetric Toda lattice equation and their algebra are constructed. An infinite class of new two-dimensional supersymmetric Toda-type hierarchies is discussed.