Integrability of Hamiltonian systems and differential Galois groups of higher variational equations

Given a complex analytical Hamiltonian system, we prove that a necessary condition for its meromorphic complete integrability is the commutativity of the identity component of the Galois group of each variational equation of arbitrary order along any integral curve. This was conjectured by the first author based on a suggestion by the third author. The first-order non-integrability criterion, obtained by the first and second authors using only first variational equations, is extended to higher orders by the present criterion. Using this result (at order two, three or higher) it is possible to solve important open problems of integrability which escaped the first order criterion.