Flat Bi-Hamiltonian Structures and Invariant Densities

A bi-Hamiltonian structure is a pair of Poisson structures P, Q which are compatible, meaning that any linear combination αP+ βQ is again a Poisson structure. A bi-Hamiltonian structure (P, Q) is called flat if P and Q can be simultaneously brought to a constant form in a neighborhood of a generic point. We prove that a generic bi-Hamiltonian structure (P, Q) on an odd-dimensional manifold is flat if and only if there exists a local density which is preserved by all vector fields Hamiltonian with respect to P, as well as by all vector fields Hamiltonian with respect to Q.