Integrability of bi-Hamiltonian systems using Casimir functions and characteristic polynomials

In this paper we prove that for a pencil of compatible Poisson brackets $\mathcal{P}=\{\mathcal{A}+\lambda \mathcal{B}\}$ the local Casimir functions of Poisson brackets $\mathcal{A}+\lambda \mathcal{B}$ and coefficients of the characteristic polynomial $p_{\mathcal{P}}$ commute w.r.t. all Poisson brackets of the pencil $\mathcal{P}$. We give a criterion when this family of functions is complete. These results generalize previous constructions of complete commutative subalgebras in the symmetric algebra $S(\mathfrak{g})$ of a finite-dimensional Lie algebra $\mathfrak{g}$ by A.S. Mishchenko & A.T. Fomenko, A.V. Bolsinov & P. Zhang and A.M. Izosimov.