  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.
