  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.
