  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.
