Some developments of the concept of Poisson manifold in the sense of A. Lichnerowicz F. Magri Dipartimento di Matematica, Universit`a di Milano T. Marsico Dottorato in Matematica, Universit` a di Milano March 5, 2015 Abstract The relationship between the theory of integrable hamiltonian sys- tems and the geometry of a special class of Poisson manifolds is dis- cussed on the ground of a concrete example, the Calogero system. Le travail est dedi` e, en respecteux hommage, ` a Mr. Andr` e Lich- nerowicz ` a l’occasion de sou 80-` eme anniversaire. 1 The Calogero systems In the middle of the sixties, Francesco Calogero drew new attention on a classical two body problem already considered by J. Bertrand a century ago: the problem of two particles interacting by force propor- tional to the cube of the inverse of the reciprocal distance. Working in a quantum context, he was able to show the quantum integrability of the three bodies problem. This result was rapidly extended in several directions. In particular Jurgens Moser, in the middle of seventies, was able to prove the complete integrability of the classical n bodies problem. The system consists of n identical particles moving on a line and re- pulsing each other by forces proportional to the cube of the mutual 1