New results concerning the so(2,1) treatment for the hypergeometric Natanzon potentials

The $so(2,1)$ analysis for the bound state sector of the hypergeometric Natanzon potentials (HNP) is extended to the scattering sector by considering the continuous series of the $so(2,1)$ algebra. As a result a complete algebraic treatment of the HNP by means of the $so(2,1)$ algebra is achieved. In the bound state sector we discuss a set of satellite potentials which arises from the action of the $so(2,1)$ generators. It is shown that the set of new potentials are not related to the one obtained by means of SUSYQM or of the potential algebra approach using the $so(2,2)$ algebra.