Motion on a given surface: Monoparametric families of orbits sufficient for separability

In the light of the inverse problem of dynamics, we study the motion of a material point on an arbitrary two-dimensional surface, submersed in E3E3. Under the assumption that a monoparametric family of geodesics and their orthogonal trajectories form an isothermic coordinate system, we prove that, if the family of geodesics is a family of orbits of the material point, compatible with the potential, then the system is integrable with an integral linear in the velocities, while, compatibility of the potential with the orthogonal trajectories guarantees integrability with a quadratic integral of motion. In both cases, we determine the form of the potential modulo one or two arbitrary functions respectively and the corresponding form of the integral, while, for the case of the orthogonal trajectories, we determine the allowed regions of motion on the surface and their stability.