  We obtain via B\"acklund 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.
