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.