Integrability conditions at order 2 for homogeneous potentials of degree -1

We prove a meromorphic integrability condition at order 2 near a homothetic orbit for a meromorphic homogeneous potential of degree -1, which extend the Morales Ramis conditions of order 1. Conversely, we prove that if this criterion is satisfied, then the Galois group of second order variational equation is abelian and we compute explicitly the Galois group and the Picard-Vessiot extension.