Local decomposition into a Kronecker-symplectic product of a bi-Hamiltonian structure (Q626069)

This paper concerns the local decomposition of a bi-Hamiltonian structure into a Kronecker-symplectic product. Let us recall that two Poisson structures \(W_1\) and \(W_2\) form a bi-Hamiltonian structure if \(W_1+W_2\) is a Poisson structure. In this paper, the author proves that, around every point of a dense open set, a bi-Hamiltonian structure decomposes into a Kronecker-symplectic product under a necessary condition of the characteristic polynomial of the symplectic factor. This result is proved in detail.