An inconvenience of all the known galoisian formulations of Ziglin's non-integrability theory is the Fuchsian condition at the singular points of the variational equations. We avoid this restriction. Moreover we prove that a necessary condition for meromorphic complete integrability (in Liouville sense) is that the identity component of the Galois group of the variational equation (in the complex domain) must be abelian. We test the efficacy of these new approaches on some examples. We will give some non academic applications in two following papers.
Given a complex analytical Hamiltonian system, we prove that a necessary condition for meromorphic complete integrability is that the identity component of the Galois group of each variational equation of arbitrary order along each integral curve must be commutative. This was conjectured by the first author based on a suggestion made by the third author due to numerical and analytical evidences concerning higher order variational equations. This non-integrability criterion extends to higher orders a non-integrability criterion (Morales-Ramis criterion), using only the first order variational equation, obtained by the first and the second author. Using our result (at order two, three or higher) it is possible to solve important open problems of integrability which escaped to Morales-Ramis criterion.
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