Self-isospectrality, special supersymmetry, and their effect on the band   structure
  We study a planar model of a non-relativistic electron in periodic magnetic and electric fields that produce a 1D crystal for two spin components separated by a half-period spacing. We fit the fields to create a self-isospectral pair of finite-gap associated Lame equations shifted for a half-period, and show that the system obtained is characterized by a new type of supersymmetry. It is a special nonlinear supersymmetry generated by three commuting integrals of motion, related to the parity-odd operator of the associated Lax pair, that coherently reflects the band structure and all its peculiarities. In the infinite period limit it provides an unusual picture of supersymmetry breaking. 
Francisco Correa, Vit Jakubsky, Luis-Miguel Nieto, and Mikhail S.   Plyushchay