  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.
