On the Superextension of the Kadomtsev-Petviashvili Equation and Finite-Gap Solutions of Korteweg-de Vries Superequations

It was shown in Ref. [1, 2], that there exist two ways for the nonlinear integrable Korteweg — de Vries equation to be super-generalized. One of this equations [1] proves to be, in fact, super-symmetric (involutory integrals of motion for this equation commute with the generator of global supertransformation), and another one [4] is bi-Hamiltonian system, i.e. the integrals of motion, which are in involution, satisfy the relations [3] 1 $$ {\Omega _1}(\delta {H_{n + 1}}) = {\Omega _2}(\delta {H_n}) $$ where Ω1 and Ω2 are the first and the second Hamiltonian structures for s-KdV superequation.