Dynamical Symmetries for Graded Vector Fields

Suppose that M = (M, A(M)) is a graded manifold and consider a direct subsheaf D of DerA(M) and a graded vector field Gamma on M; both satisfying certain conditions. We associate to the graded vector field Gamma is an element of DerA(M), a set of 1-forms (DerA(M))*(Gamma) and show that if phi is an element of (DerA(M))*(Gamma) is a non-degenerate graded 1-form and X is an element of DerA(M) such that for some superfunction f on M, L-X(J(phi)) = df, and i(X) (phi - (-1)(vertical bar Gamma vertical bar(vertical bar phi vertical bar+1)) d(phi(Delta))) = 0, then the superfunction F = f - J(phi)(X) satisfies L Gamma F = Gamma(F) = 0. This result, generalizes the conditions under which there exist a solution for the inverse problem.