Calogero-Moser

Consider n classical particles on a line that repel each other with a inverse square law. You would not expect this to be completely integrable. But it's this presentation of the phase space that is hiding the many other Hamiltonians available to you. You just don't see them; I don't blame you.

So how should you think of this system?

Start with the cotangent bundle of the space n by n square matrices. Identify it as the space of pairs of n by n matrices (X,Y). Then perform a symplectic reduction  with moment map given by the commutator of this pair which maps to the lie algebra sl(n,C). But instead of the inverse image of 0 which would give you commuting pairs (the commuting scheme), take the inverse image of the set of traceless matrices T such that T+1 has rank 1.

Taking the eigenvalues of X as the positions of the n particles. You will find the off diagonal terms of Y will be determined, but the diagonals won't be. Identify these diagonal entries as the momenta. If you take the function on this space Tr(Y^2) you end up getting the Hamiltonian that we originally said.

But now we see the other Hamiltonians too. They are Tr(Y^k). This gives the rational Calogero-Moser. You can choose another system of Hamiltonians like Tr((XY)^i) for another integrable system. The one just said was the trigonometric one. It is so called because the force between the particles goes like the csc(x) of the displacement.

For more details, see

http://www-math.mit.edu/~etingof/zlecnew.pdf