Meromorphic parametric non-integrability; the inverse square potential

We let H(X, Y, alpha) be a Hamiltonian depending meromorphically on positions X, inertial momenta Y and parameters ct. In Theorem 1 we give conditions for the "meromorphic parametric" non-integrability of H. Theorem 2 proves the meromorphic non-integrability of the 4-body problem on a line with given masses (1, m, m, 1) with m not equal 1, and of the 3-body problem in P-p with p greater than or equal to 2 and given masses (1, 1, m), for the inverse square potential. Those are the simplest cases left open after the integrability results of Jacobi (3 bodies on a line with arbitrary masses) and Calogero-Moser (n bodies on a line with equal masses). Taking the masses as parameters and using both Theorems 1 and 2, we prove Theorem 3, which shows meromorphic parametric non-integrability results for the inverse square potential.