Stability in bi-Hamiltonian systems and multidimensional rigid body

The presence of two compatible Hamiltonian structures is known to be one of the main, and the most natural, mechanisms of integrability. For every pair of Hamiltonian structures, there are associated conservation laws (first integrals). Another approach is to consider the second Hamiltonian structure on its own as a tensor conservation law. The latter is more intrinsic as compared to scalar conservation laws derived from it and, as a rule, it is "simpler". Thus it is natural to ask: can the dynamics of a bi-Hamiltonian system be understood by studying its Hamiltonian pair, without studying the associated first integrals?\par In this paper, the problem of stability of equilibria in bi-Hamiltonian systems is considered and it is shown that the conditions for nonlinear stability can be expressed in algebraic terms of linearization of the underlying Poisson pencil. This is used to study stability of stationary rotations of a free multidimensional rigid body.