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Extended supersymmetry of the self-isospectral crystalline and soliton chains

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.