Bi-Hamiltonian structures with symmetries, Lie pencils and integrable systems

There are two classical ways of constructing integrable systems by means of bi-Hamiltonian structures. The first one supposes nondegeneracy of one of the Poisson structures generating the pencil and uses the so-called recursion operator. This situation corresponds to the absence of Kronecker blocks in the so-called Jordan-Kronecker decomposition. The second one, which corresponds to the absence of Jordan blocks in this decomposition, uses the Casimir functions of different members of the pencil. In this paper, we consider the general case of a bi-Hamiltonian structure with both Kronecker and Jordan blocks and give a criterion for the completeness of the corresponding family of functions. This result is related to a natural action of some Lie algebra which gives a symmetry of the whole pencil. The criterion is applied to bi-Hamiltonian structures related to Lie pencils, although we also discuss other possible applications.