Peculiarities of the hidden nonlinear supersymmetry of the Poschl-Teller system in the light of the Lame equation

A hidden nonlinear bosonized supersymmetry was revealed recently in the Poschl-Teller and finite-gap Lame systems. In spite of the intimate relationship between the two quantum models, the hidden supersymmetry in them displays essential differences. In particular, the kernel of the supercharges of the Poschl-Teller system, unlike the case of the Lame equation, includes nonphysical states. By means of the Lame equation, we clarify the nature of these peculiar states, and show that they encode essential information not only on the original hyperbolic Poschl-Teller system, but also on its singular hyperbolic and trigonometric modifications, and reflect the intimate relation of the model to a free-particle system.