Integrability of Hamiltonian systems with homogeneous potentials of degree zero

We derive necessary conditions for integrability in the Liouville sense of classical Hamiltonian systems with homogeneous potentials of degree zero. We obtain these conditions through an analysis of the differential Galois group of variational equations along a particular solution generated by a non-zero solution d∈Cnd∈Cn of nonlinear equation gradV(d)=dgradV(d)=d. We prove that when the system is integrable the Hessian matrix V″(d)V″(d) has only integer eigenvalues and is diagonalizable.