Hamiltonian dynamical systems with even and odd Poisson brackets

Using the example of the supersymmetric Witten mechanics we prove that the Hamiltonian systems with equal number of Grassmann even and Grassmann odd canonical variables are inherently characterized by the odd Poisson bracket in addition to the even one. The duality between even and odd integrals of motion under the change of the Poisson brackets parity is established for such systems.