Lax Integrable Supersymmetric Hierarchies on Extented Phase Spaces

We obtain via Bäcklund transformation the Hamiltonian representation for a Lax type nonlinear dynamical system hierarchy on a dual space to the Lie algebra of super-integral-differential operators of one anticommuting variable, extended by evolutions of the corresponding spectral problem eigenfunctions and adjoint eigenfunctions, as well as for the hierarchies of their additional symmetries. The relation of these hierarchies with the integrable by Lax ($2|1+1$)-dimensional supersymmetric Davey–Stewartson system is investigated.