Isospectrality of conventional and new extended potentials, second-order   supersymmetry and role of PT symmetry
  We develop a systematic approach to construct novel completely solvable rational potentials. Second-order supersymmetric quantum mechanics dictates the latter to be isospectral to some well-studied quantum systems. $\cal PT$ symmetry may facilitate reconciling our approach to the requirement that the rationally-extended potentials be singularity free. Some examples are shown. 
B. Bagchi, C. Quesne, R. Roychoudhury