A canonical basis of a pair of compatible Poisson brackets on a matrix algebra

Given an arbitrary complex matrix $A$ and a generic matrix $X$ we find a canonical basis for the Kronecker part of the bi-Lagrangian subspace with respect to the corresponding Poisson brackets on the Lie algebra $\mathfrak{gl}_n(\mathbb C)$, and also find a system of functions in bi-involution corresponding to this basis. In particular, for nilpotent matrices $A$ we prove that all nonzero functions obtained by applying the Mishchenko-Fomenko argument shift method to the coefficients of the characteristic polynomial form the Kronecker part of the complete system of functions in bi-involution.
Bibliography: 9 titles.