On the geometry of the phase spaces of some SO(2, 1) invariant systems

Some SO(2, 1) invariant systems, like the Pöschl-Teller potential (both trigonometric and hyperbolic) and the Morse potential are studied by means of a transformation relating their phase-spaces with the coadjoint orbits of SO(2, 1) or SO(3). The geometries of these phase spaces are studied and their effects on the quantization are commented.