Existence of different intermediate Hamiltonians in type A NN-fold supersymmetry II: The NN= 3 case

We continue the previous study on the existence of different intermediate Hamiltonians in type A NN-fold supersymmetric systems and carry out an exhaustive investigation on the NN = 3 case. In contrast with the NN = 2 case, we find various patterns in the existence of intermediate Hamiltonians due to the presence of two different intermediate positions in a factorized type A 3-fold supercharge. In addition, all the NN = 3 models are strictly restricted to at most elliptic type, which enables us to make the complete classification of the systems which admit intermediate Hamiltonians. Finally, we show realizations of third-order parasupersymmetry and variant generalized 3-fold superalgebras by such systems.