Note on integrability of certain homogeneous Hamiltonian systems

•Necessary integrability conditions for some 2D homogeneous Hamilton systems are given.•Conditions are obtained analysing differential Galois group of variational equations.•New integrable and superintegrable systems are identified.

In this paper we investigate a class of natural Hamiltonian systems with two degrees of freedom. The kinetic energy depends on coordinates but the system is homogeneous. Thanks to this property it admits, in a general case, a particular solution. Using this solution we derive necessary conditions for the integrability of such systems investigating differential Galois group of variational equations.