Equivalence between Type I Liouville dynamical systems in the plane and the sphere

Separable Hamiltonian systems either in sphero-conical coordinates on a $S^2$ sphere or in elliptic coordinates on a ${mathbb R}^2$ plane are described in an unified way. A back and forth route connecting these Liouville Type I separable systems is unveiled. It is shown how the gnomonic projection and its inverse map allow us to pass from a Liouville Type I separable system with an spherical configuration space to its Liouville Type I partner where the configuration space is a plane and back. Several selected spherical separable systems and their planar cousins are discussed in a classical context.