  Motivated by the concept of shape invariance in supersymmetric quantum
mechanics, we obtain potentials whose spectrum consists of two shifted sets of
equally spaced energy levels. These potentials are similar to the
Calogero-Sutherland model except the singular term $\alpha x^{-2}$ always falls
in the transition region $-1/4 < \alpha < 3/4$ and there is a delta-function
singularity at x=0.
