Everyone of us had sometimes this awful feeling that some sign is lost in a calculation and that this sign is perturbing some fundamental understanding of what is going on. I feel the same has happened for me today and I can't figure this sign problem out, so I count on you.

A Calogero-Moser system is defined as a Hamiltonian system with a Hamiltonian $H=\sum p_i^2 + \sum_{i \neq k} \frac{1}{x_i-x_j}^2$. It is widely known that this system is completely integrable.

I am trying to understand this widely known fact.

One of the proofs relies on the relation of tho system with a linear flow in the space of matrices, the relationship is nicely explained in this Mathoverflow entrance: Is the 'massive' Calogero-Moser system still integrable?

The question that I already asked as a comment there is the following: the standard proof of the integrability rewrites $H$ as a restriction of some other function on the space of matrices which is actually $\mathrm{Tr }Y^2$ for a matrix $Y$ defined by $$ Y_{ii}=p_i, \; Y_{ik}=(x_i-x_k)^{-1}, \; i\neq k $$. A simple calculation will give us not $H$ but

$H^-=\sum p_i^2 - \sum_{i \neq k} \frac{1}{x_i-x_j}^2$

This is exactly the expression Etingof obtains in his Lectures on Calogero-Moser systems. http://www-math.mit.edu/~etingof/zlecnew.pdf Etingof starts from the dynamics on matrix space and defines CM system as its symplectic reduction. So no problems for him.

But for a system of the particles on the real line, I feel lost. How one can prove integrability? And also, $H^-$ is giving the trajectories that would collapse.