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

$$\begin{equation*} H=\frac{1}{2}{\boldsymbol p}^T{\boldsymbol M}{\boldsymbol p} +V({\boldsymbol q}), \end{equation*}$$

where ${\boldsymbol q}=(q_1, \ldots, q_n)\in\mathbb C^n$, ${\boldsymbol p}=(p_1, \ldots, p_n)\in\mathbb C^n$, are the canonical coordinates and momenta, $\boldsymbol M$ is a symmetric non-singular matrix, and $V({\boldsymbol q})$ is a homogeneous function of degree $k\in\mathbb Z^{\star}$. 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.