  Some time ago, Thies et al. showed that the Gross-Neveu model with a bare
mass term possesses a kink-antikink crystalline phase. Corresponding
self-consistent solutions, known earlier in polymer physics, are described by a
self-isospectral pair of one-gap periodic Lame potentials with a Darboux
displacement depending on the bare mass. We study an unusual supersymmetry of
such a second order Lame system, and show that the associated first order
Bogoliubov-de Gennes Hamiltonian possesses the own nonlinear supersymmetry. The
Witten index is ascertained to be zero for both of the related exotic
supersymmetric structures, each of which admits several alternatives for the
choice of a grading operator. A restoration of the discrete chiral symmetry at
zero value of the bare mass, when the kink-antikink crystalline condensate
transforms into the kink crystal, is shown to be accompanied by structural
changes in both of the supersymmetries. We find that the infinite period limit
may or may not change the index. We also explain the origin of the Darboux
dressing phenomenon recently observed in a non-periodic self-isospectral
one-gap Poschl-Teller system, which describes kink-antikink baryons of Dashen,
Hasslacher and Neveu.
