Bäcklund transformations for the difference Hirota equation and the supersymmetric Bethe ansatz

We consider $GL(K\mid M)$-invariant integrable supersymmetric spin chains with twisted boundary conditions and demonstrate the role of Bäcklund transformations in solving the difference Hirota equation for eigenvalues of their transfer matrices. We show that the nested Bethe ansatz technique is equivalent to a chain of successive Bäcklund transformations "undressing" the original problem to a trivial one.