Integrals in Involution for Groups of Linear Symplectic Transformations and Natural Mechanical Systems with Homogeneous Potential

We prove that if a complex Hamiltonian system with $n$ degrees of freedom has $n$ functionally independent meromorphic first integrals in involution and the monodromy group of the corresponding variational system along some phase curve has $n$ pairwise skew-orthogonal two-dimensional invariant subspaces, then the restriction of the action of this group to each of these subspaces has a rational first integral. The result thus obtained is applied to natural mechanical systems with homogeneous potential, in particular, to the $n$-body problem.