Integrability analysis of natural Hamiltonian systems in curved spaces

In this work we perform integrability analysis of natural Hamiltonian systems with two degrees of freedom governed by a metric defining the infinitesimal linear element dl2=12[a(r)−1dr2+b(r)−1dφ2], where a(r) and b(r) are arbitrary meromorphic functions of variable r. We formulate necessary integrability conditions for such systems in a framework of differential Galois theory. We present several examples which show that the obtained conditions are simple for applications and effective.