Is the 'massive' Calogero-Moser system still integrable?

The paper "Meromorphic Parametric Non-Integrability, the Inverse Square Potential" by E. J. Tosel, proves almost what was claimed in the comments. Except for Jacobi's theorem:

The 3-body problem on a line with arbitrary masses and inverse square potential is completely integrable with rational first integrals.

and Calogero-Moser’s Theorem:

The n-body problem with equal masses on a line with an inverse square potential is completely integrable. More precisely, there exists a complete family of commuting first integrals which are rational in $(Q,P)$.

all other cases are non-integrable. The main theorem is:

Theorem 3 (Non-integrability meromorphic in linear momenta and masses, rational in positions).

(i) For $n = 4$, the $n$-body problem on a line with an inverse square potential does not have a complete system of generically independent first integrals which are rational with respect to $Q$ and meromorphic with respect to $P$ and $(m _i) _{1\le i\le n}$

(ii) For $n = 3$ and $p \geq 2$, the $n$-body problem in $\mathbb{R} ^p$ with an inverse square potential does not have a complete system of generically independent first integrals which are rational with respect to $Q$ and meromorphic with respect to $P$ and $(m _i) _{1\le i\le n}$

A corollary of this theorem is that Calogero-Moser’s theorem deals with an exceptional case: there cannot exist a Lax pair $(L,B)$ which would depend meromorphically on the masses for $n$ bodies on a line ($n \geq 4$).

There is a remark there that the rationality condition in the case of the line needs only be checked in $n-4$ positions.

Just to add to Gjergji Zaimi's answer, Harry Braden has sent me the expressions for the conserved charges responsible for the integrability of the $N=3$ model:

• The total momentum $P = p_1 + p_2 + p_3$

• The hamiltonian $H$

• $Q = 2 H \sum_{i=1}^3 m_i x_i^2 -(\sum_{i=1}^3 x_i p_i )^2$