We characterize and completely describe some types of separable potentials in two-dimensional spaces, , of any (positive, zero or negative) constant curvature and either definite or indefinite signature type. The results are formulated in a way which applies at once for the two-dimensional sphere S2, hyperbolic plane H2, AntiDeSitter / DeSitter two-dimensional spaces AdS1+1 / dS1+1 as well as for their flat analogues E2 and M1+1. This is achieved through an approach of Cayley-Klein type with two parameters, κ1 and κ2, to encompass all curvatures and signature types. We discuss six coordinate systems allowing separation of the Hamilton-Jacobi equation for natural Hamiltonians in and relate them by a formal triality transformation, which seems to be a clue to introduce general “elliptic coordinates” for any CK space concisely. As an application we give, in any , the explicit expressions for the Fradkin tensor and for the Runge-Lenz vector, i.e., the constants of motion for the harmonic oscillator and Kepler potential on any .