We present various properties of algebraic potentials, and then prove that some Morales-Ramis theorems readily apply for such potentials even if they are not in general meromorphic potentials. This allows in particular to precise some non-integrability proofs in celestial mechanics, where the mutual distances between the bodies appear in the potentials, and thus making this analysis unavoidable.

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 variational equation is abelian and we compute explicitly the Galois group and the Picard-Vessiot extension.

We prove an integrability criterion and a partial integrability criterion for homogeneous potentials of degree −1 which are invariant by rotation. We then apply it to the proof of the meromorphic non-integrability of the n body problem with Newtonian interaction in the plane on a surface of equation (H, C) = (H 0 , C 0 ) with (H 0 , C 0 ) = (0, 0) where C is the angular momentum and H the energy, in the case where the n masses are equal.

Abstract. We prove an integrability criterion of order 3 for a homogeneous potential of degree −1 in the plane. Still, this criterion depends on some integer and it is impossible to apply it directly except for families of potentials whose eigenvalues are bounded. To address this issue, we use holonomic and asymptotic computations with error control of this criterion and apply it to the potential of the form V (r, θ) = r −1 h(exp(iθ)) with h ∈ C[z], deg h ≤ 3. We find then all meromorphically integrable potentials of this form.