Singularities of bihamiltonian systems and the multidimensional rigid body

Two Poisson brackets are called compatible if any linear combination of these brackets is a Poisson bracket again. The set of non-zero linear combinations of two compatible Poisson brackets is called a Poisson pencil. A system is called bihamiltonian (with respect to a given pencil) if it is hamiltonian with respect to any bracket of the pencil. The property of being bihamiltonian is closely related to integrability. On the one hand, many integrable systems known from physics and geometry possess a bihamiltonian structure. On the other hand, if we have a bihamiltonian system, then the Casimir functions of the brackets of the pencil are commuting integrals of the system. We consider the situation when these integrals are enough for complete integrability. As it was shown by Bolsinov and Oshemkov, many properties of the system in this case can be deduced from the properties of the Poisson pencil itself, without explicit analysis of the integrals. Developing these ideas, we introduce a notion of linearization of a Poisson pencil. In terms of linearization, we give a criterion for non-degeneracy of a singular point and describe its type. These results are applied to solve the stability problem for a free multidimensional rigid body.