Jordan obstruction to the integrability of Hamiltonian systems with homogeneous potentials

In this paper, we consider the natural complex Hamiltonian systems with homogeneous potential $V(q)$, $q\in {\text{C}}^n$, of degree $k\in {\mathbb{Z}}^{\star }$. The known results of Morales and Ramis give necessary conditions for the complete integrability of such systems. These conditions are expressed in terms of the eigenvalues of the Hessian matrix $V\mathrm{"}(c)$ calculated at a non-zero point $c\in {\text{C}}^n$, such that $V^{\mathrm{\prime }}(c)=c$. The main aim of this paper is to show that there are other obstructions for the integrability which appear if the matrix $V\mathrm{"}(c)$ is not diagonalizable. We prove, among other things, that if $V\mathrm{"}(c)$ contains a Jordan block of size greater than two, then the system is not integrable in the Liouville sense. The main ingredient in the proof of this result consists in translating some ideas of Kronecker about Abelian extensions of number fields into the framework of differential Galois theory.