Necessary conditions for the existence of additional first integrals for Hamiltonian systems with homogeneous potential

We consider a natural Hamiltonian system of n degrees of freedom with a homogeneous potential. We assume that the system admits 1 <= m < n independent and commuting first integrals F-1, ... F-m. We give easily computable and effective necessary conditions for the existence of one additional first integral Fm+1 such that all integrals F-1,... Fm+1 are independent, pairwise commute and are meromorphic in a connected neighbourhood of a certain phase curve. These conditions are obtained from an analysis of the differential Galois group of variational equations along a particular solution of the system. We apply our result analysing the problem of the existence of one additional first integral for a homogeneous nonlinear lattice on a line.