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We report our recent results concerning integrability of Hamiltonian systems governed by Hamilton’s function of the form H = 1 2 Pn i=1 p 2 i +V (q), where the potential V is a finite sum of homogeneous components. In this paper we show how to find, in the differential Galois framework, computable necessary conditions for the integrability of such systems. Our main result concerns potentials of the form V = Vk + VK , where Vk and VK are homogeneous functions of integer degrees k and K > k, respectively. We present examples of integrable systems which were obtained by applying our main theorem.

Overview of the differential Galois integrability conditions for non-homogeneous potentials