  We study supersymmetry of a self-isospectral one-gap Poschl-Teller system in
the light of a mirror symmetry that is based on spatial and shift reflections.
The revealed exotic, partially broken nonlinear supersymmetry admits seven
alternatives for a grading operator. One of its local, first order supercharges
may be identified as a Hamiltonian of an associated one-gap, non-periodic
Bogoliubov-de Gennes system. The latter possesses a nonlinear supersymmetric
structure, in which any of the three non-local generators of a Clifford algebra
may be chosen as the grading operator. We find that the supersymmetry
generators for the both systems are the Darboux-dressed integrals of a free
spin-1/2 particle in the Schrodinger picture, or of a free massive Dirac
particle. Nonlocal Foldy- Wouthuysen transformations are shown to be involved
in the supersymmetric structure.
