Commutative families of functions related to consistent Poisson brackets (Q1181704)

Bi-Hamiltonian systems are considered, i.e. dynamical systems that are Hamiltonian with respect to two compatible Poisson structures. Under certain conditions such systems possess a number of integrals, mutually in involution. The author restricts himself to the case of finite- dimensional phase space and investigates the problem of completeness of the mentioned family of integrals in the Liouville sense. Sufficient conditions of completeness are presented. A specification of the general approach is given, when the pair of compatible Poisson structures is obtained by the so called argument shift method on manifolds being finite dimensional Lie coalgebras.