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We investigate a peculiar supersymmetry of the pairs of reflectionlessquantum mechanical systems described by n-soliton potentials of a general formthat depends on n scaling and n translation parameters. We show that if all thediscrete energy levels of the subsystems are different, the superalgebra, beinginsensitive to translation parameters, is generated by two supercharges ofdifferential order 2n, two supercharges of order 2n+1, and two bosonicintegrals of order 2n+1 composed from Lax integrals of the partners. The exoticsupersymmetry undergoes a reduction when r discrete energy levels of onesubsystem coincide with any r discrete levels of the partner, the total orderof the two independent intertwining generators reduces then to 4n-2r+1, and thenonlinear superalgebraic structure acquires a dependence on r relativetranslations. For a complete pairwise coincidence of the scaling parameterswhich control the energies of the bound states and the transmission scatteringamplitudes, the emerging isospectrality is detected by a transmutation of oneof the Lax integrals into a bosonic central charge. Within the isospectralclass, we reveal a special case giving a new family of finite-gap first orderBogoliubov-de Gennes systems related to the AKNS integrable hierarchy.

Effect of scalings and translations on the supersymmetric quantum mechanical structure of soliton systems