  We study supersymmetric structure of the self-isospectral crystalline chains
formed by N copies of the mutually displaced one-gap Lame systems. It is
generated by the N(N-1) integrals of motion which are the first order matrix
differential operators, by the same number of the nontrivial second order
integrals, and by the N third order Lax integrals. We show that the structure
admits distinct alternatives for a grading operator, and in dependence on its
choice one of the third order matrix integrals plays either the role of the
bosonic central charge or the role of the fermionic supercharge to be a square
root of the spectral polynomial. Yet another peculiarity is that the set of all
the second order integrals of motion generates a nonlinear sub-superalgebra. We
also investigate the associated self-isospectral soliton chains, and discuss
possible physical applications of the unusual extended supersymmetry.
