Recursion operators and constants of motion in supermechanics (Q1295491)

We prove that only even graded Poisson brackets can be characterized by the vanishing of the graded Schouten bracket of the associated graded tensor of type \((0,2)\) with itself. On the other hand, we prove that the supertraces of different powers of an invariant graded tensor of type \((1,1)\) are constants of motion, and that when such tensor is even, the infinitesimal supersymmetries it generates out of a given infinitesimal supersymmetry form a supercommutative superalgebra.