Existence of different intermediate Hamiltonians in type A NN-fold supersymmetry

Type A NN-fold supercharge admits a one-parameter family of factorizations into product of NN first-order linear differential operators due to an underlying GL(2,C) symmetry. As a consequence, a type A NN-fold supersymmetric system can have different intermediate Hamiltonians corresponding to different factorizations. We derive the necessary and sufficient conditions for the latter system to possess intermediate Hamiltonians for the N=2N=2 case. We then show that whenever it has (at least) one intermediate Hamiltonian, it can admit second-order parasupersymmetry and a generalized 2-fold superalgebra. As an illustration, we construct a set of generalized Pöschl–Teller potentials of this kind.