  In this paper, we discuss the parametric symmetries in different exactly
solvable systems characterized by real or complex P T symmetric potentials. We
focus our at- tention on the conventional potentials such as the generalized
Poschl Teller (GPT), o Scarf-I and P T symmetric Scarf-II which are invariant
under certain parametric transformations. The resulting set of potentials are
shown to yield a completely dif- ferent behavior of the bound state solutions.
Further the supersymmetric (SUSY) partner potentials acquire different forms
under such parametric transformations leading to new sets of exactly solvable
real and P T symmetric complex potentials. These potentials are also observed
to be shape invariant (SI) in nature. We subse- quently take up a study of the
newly discovered rationally extended SI Potentials, corresponding to the above
mentioned conventional potentials, whose bound state solutions are associated
with the exceptional orthogonal polynomials (EOPs). We discuss the
transformations of the corresponding Casimir operator employing the properties
of the so(2,1) algebra.
