Super-integrable 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 the most interesting one. This dynamical system is 2 N -dimensional with 2 N −1 first integrals and our construction yields 2 N −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.