Bi-Hamiltonian partially integrable systems

Given a first order dynamical system possessing a commutative algebra of dynamical symmetries, we show that, under certain conditions, there exists a Poisson structure on an open neighborhood of its regular (not necessarily compact) invariant manifold which makes this dynamical system into a partially integrable Hamiltonian system. This Poisson structure is by no means unique. Bi-Hamiltonian partially integrable systems are described in some detail. As an outcome, we state the conditions of quasiperiodic stability (the KAM theorem) for partially integrable Hamiltonian systems. (C) 2003 American Institute of Physics.