On the integrability of 2D Hamiltonian systems with variable Gaussian curvature

In this work, we consider the integrability of a general 2D motion of a particle on a surface with variable Gaussian curvature under the influence of conservative potential forces. Although this system has a kinetic energy relying on the coordinates, it remains homogeneous. The homogeneity of the system generally enables us to find a particular solution that can be utilized to derive the necessary conditions for the integrability by studying the properties of the differential Galois group of the normal variational equations along this particular solution. We present a new theory that can be applied to determine the necessary conditions for the integrability of Hamiltonian systems having a variable Gaussian curvature.