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. $\mathcal{P}\mathcal{T}$ symmetry may facilitate reconciling our approach to the requirement that the rationally-extended potentials be singularity free. Some examples are shown.