Local bi-integrability of bi-Hamiltonian systems via bi-Poisson reduction

Abstract

We prove that any bi-Hamiltonian system $v=\left(\mathcal{A}+\lambda \mathcal{B}\right)d{H}_{\lambda }$ that is Hamiltonian with respect all Poisson brackets $\mathcal{A}+\lambda \mathcal{B}$ is locally bi-integrable in both the real smooth case, when all eigenvalues of the Poisson pencil $\mathcal{P}=\left\{\mathcal{A}+\lambda \mathcal{B}\right\}$ are real, and in the complex analytic case. A complete set of functions in bi-involution is constructed by extending the set of standard integrals, which consists of Casimir functions of Poisson brackets, eigenvalues of the Poisson pencil and Hamiltonians.