  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.
