Superintegrable Calogero-type systems admit maximal number of Poisson structures

Abstract

We present a general scheme for constructing the Poisson structure of super-integrable dynamical systems of which the rational Calogero-Moser system is one of the most interesting one. This dynamical system is $2N$ dimensional with $2N- 1$ first integrals and our construction yields $2N-1$ degenerate Poisson tensors that each admit $2(N-1)$ Casimirs. Our results are quite generally applicable to all super-integrable systems and form an alternative to the traditional bi-Hamiltonian approach.