Shape Invariant Natanzon Potentials from Potential Algebra
  Using the underlying algebraic structures of Natanzon potentials, we discuss conditions that generate shape invariant potentials. In fact, these conditions give all the known shape invariant potentials corresponding to a translational change of parameters. We also find that while the algebra for the general Natanzon potential is $SO(2,2)$, a subgroup $SO(2,1)$ suffices for all the shape invariant problems of Natanzon type. 
Asim Gangopadhyayaa, Jeffry V. Mallow and Uday P. Sukhatme