Partial integrability of Hamiltonian systems with homogeneous potential

In this paper we consider systems with n degrees of freedom given by the natural Hamiltonian function of the form
$H = \frac{1}{2} {\bf p}^T {\bf Mp} + V({\bf q})$,
where ${\bf q} = (q_1, \ldots, q_n) \in \mathbb{C}^n$, ${\bf p}= (p_1, \ldots, p_n) \in \mathbb{C}^n$, are the canonical coordinates and momenta, $\bf M$ is a symmetric non-singular matrix, and $V({\bf q})$ is a homogeneous function of degree $k \in Z^*$. We assume that the system admits $1 \leqslant m < n$ independent and commuting first integrals $F_1, \ldots F_m$. Our main results give easily computable and effective necessary conditions for the existence of one more additional first integral $F_{m+1}$ such that all integrals $F_1, \ldots F_{m+1}$ are independent and pairwise commute. These conditions are derived from an analysis of the differential Galois group of variational equations along a particular solution of the system. We apply our result analysing the partial integrability of a certain $n$ body problem on a line and the planar three body problem.