Flat bi-Hamiltonian structures and invariant densities

A bi-Hamiltonian structure is a pair of Poisson structures $\mathcal P$, $\mathcal Q$ which are compatible, meaning that any linear combination $\alpha \mathcal P + \beta \mathcal Q$ is again a Poisson structure. A bi-Hamiltonian structure $(\mathcal P, \mathcal Q)$ is called flat if $\mathcal P$ and $\mathcal Q$ can be simultaneously brought to a constant form in a neighborhood of a generic point. We prove that a generic bi-Hamiltonian structure $(\mathcal P, \mathcal Q)$ 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 $\mathcal P$, as well as by all vector fields Hamiltonian with respect to $\mathcal Q$.