The Lax integrable supersymmetric hierarchies on extended phase spaces
Since the paper of M. Adler  there was understood that a wide class of Lax type integrable Korteweg-de Vries nonlinear systems and their supersymmetric analogs [2, 3] of one and two anticommuting variables could be considered as Hamiltonian flows, generated by the R-deformed canonical Lie-Poisson structure and Casimir functionals, on a dual space to the Lie algebra of integral-differential operators. The existence of Hamiltonian representation for these flows, added by correponding evolutions of associated spectral problem eigenfunctions and adjoint eigenfunctions, in the case of super-integro-differential operators of one and two anticommuting variables is investigated by use of the invariant Casimir functionals' property under some Lie-Backlund transformation .
The following hierarchies of additional symmetries in a Lax type form
are proven to be Hamiltonian ones. It is shown that the additional symmetry
are generated by the Poisson structure, being a tensor product of the canonical
Lie-Poisson and some canonical finite-dimensional super-Poisson ones, and
a functional, being a sum of Casimir one and the corresponding power of
a spectral eigenvalue. The connection of additional symmetry hierarchies
with (2|1+1)- and (2|2+1)-dimensional supersymmetric Davey-Stewartson equations
of one and two anticommuting variables accordingly and their triple linear
representations are established.