Integrability of supersymmetric Calogero-Moser models

We analyze the integrability of the ${\cal N}$-extended supersymmetric Calogero-Moser model. We explicitly construct the Lax pair $\{L,A\}$ for this system, which properly reproduces all equations of motion. After adding a supersymmetric oscillator potential we reduce the latter to solving $\dot{U}\,{=}\,A\,U$ for the time evolution operator $U(t)$. The bosonic variables, however, evolve independently of $U$ on closed trajectories, as is required for superintegrability. To visualize the structure of the conserved currents we derive the complete set of Liouville charges up to the fifth power in the momenta, for the ${\cal N}{=}\,2$ supersymmetric model. The additional, non-involutive, conserved charges needed for a maximal superintegrability of this model are also found.