Integrable spin-boson interaction in the Tavis-Cummings model from a generic boundary twist

We construct models describing interaction between a spin s and a single bosonic mode using a quantum inverse scattering procedure. The boundary conditions are generically twisted by generic matrices with both diagonal and off-diagonal entries. The exact solution is obtained by mapping the transfer matrix of the spin-boson system to an auxiliary problem of a spin-j coupled to the spin-s with general twist of the boundary condition. The corresponding auxiliary transfer matrix is diagonalized by a variation of the method of Q-matrices of Baxter. The exact solution of our problem is obtained applying certain large-j limit to su( 2)(j), transforming it into the bosonic algebra.