TITLE Integrability of Hamiltonian Systems and Differential Galois Groups of Higher Variational Equations AUTHORS: Juan J. Morales-Ruiz^(1), Jean-Pierre Ramis^(2) and Carles Sim\'o^(3) (1) Departament de Matem\`atica Aplicada II, Universitat Polit\`ecnica de Catalunya, Edifici Omega, Campus Nord c/ Jordi Girona, 1-3, E-08034 Barcelona, Spain (2) Institut Universitaire de France and Laboratoire Emile Picard Universit\'e Paul Sabatier. 118, route de Narbonne, Toulouse, Cedex, France (3) Departament de Matem\`atica Aplicada i An\`alisi, Univ. de Barcelona Gran Via 585, E-08007 Barcelona, Spain E-mail: Juan.Morales-Ruiz@upc.es, ramis@picard.ups-tlse.fr, carles@maia.ub.es ABSTRACT ven 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.