Rigidity, Functional Equations and the Calogero-Moser Model

Suppose we have a natural Hamiltonian $H$ of $n$ particles on the line, centre of mass momentum $P$ and a further independent quantity $Q$, cubic in the momenta. If these are each $S_n$ invariant and mutually Poisson commute we have the Calogero-Moser system with potential $V={1/6}\sum\limits_{i\neq j}\wp(q_{i}-q_{j}) +const$.