  We show in detail how Richardson's exact solution of a discrete-state BCS
(DBCS) model can be recovered as a special case of an algebraic Bethe Ansatz
solution of the inhomogeneous XXX vertex model with twisted boundary
conditions: by implementing the twist using Sklyanin's K-matrix construction
and taking the quasiclassical limit, one obtains a complete set of conserved
quantities, H_i, from which the DBCS Hamiltonian can be constructed as a second
order polynomial. The eigenvalues and eigenstates of the H_i (which reduce to
the Gaudin Hamiltonians in the limit of infinitely strong coupling) are exactly
known in terms of a set of parameters determined by a set of on-shell Bethe
Ansatz equations, which reproduce Richardson's equations for these parameters.
We thus clarify that the integrability of the DBCS model is a special case of
the integrability of the twisted inhomogeneous XXX vertex model. Furthermore,
by considering the twisted inhomogeneous XXZ model and/or choosing a generic
polynomial of the H_i as Hamiltonian, more general exactly solvable models can
be constructed. -- To make the paper accessible to readers that are not Bethe
Ansatz experts, the introductory sections include a self-contained review of
those of its feature which are needed here.
