Analytic integrability for a class of cone potential Hamiltonian systems

The main purpose of the present paper is to prove the analytic Liouville integrability of some classes of Hamiltonian systems, using the components of the asymptotic velocities as the constants of motion, independent and in involution. These classes include, for example, a system of n particles on the line, interacting under some potentials like the inverse r-power potential with r > 0 (including the Coulombian potential, r = 1, and the Calogero-Moser potential, r = 2). The method can be adapted to the Toda-like case and in particular to the non periodic Toda System.