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In this paper, we consider the natural complex Hamiltonian systems withhomogeneous potential $V(q)$, $q\in\C^n$, of degree $k\in\Z^\star$. The knownresults of Morales and Ramis give necessary conditions for the completeintegrability of such systems. These conditions are expressed in terms of theeigenvalues of the Hessian matrix $V"(c)$ calculated at a non-zero point$c\in\C^n$, such that $V'(c)=c$. The main aim of this paper is to show thatthere are other obstructions for the integrability which appear if the matrix$V"(c)$ is not diagonalizable. We prove, among other things, that if $V"(c)$contains a Jordan block of size greater than two, then the system is notintegrable in the Liouville sense. The main ingredient in the proof of thisresult consists in translating some ideas of Kronecker about Abelian extensionsof number fields into the framework of differential Galois theory.

Jordan obstruction to the integrability of Hamiltonian systems with homogeneous potentials