  We construct commuting transfer matrices for models describing the
interaction between a single quantum spin and a single bosonic mode using the
quantum inverse scattering framework. The transfer matrices are obtained from
certain inhomogeneous rational vertex models combining bosonic and spin
representations of SU(2), subject to non-diagonal toroidal and open boundary
conditions. Only open boundary conditions are found to lead to integrable
Hamiltonians combining both rotating and counter-rotating terms in the
interaction. If the boundary matrices can be brought to triangular form
simultaneously, the spectrum of the model can be obtained by means of the
algebraic Bethe ansatz after a suitable gauge transformation; the corresponding
Hamiltonians are found to be non-hermitian. Alternatively, a certain
quasi-classical limit of the transfer matrix is considered where hermitian
Hamiltonians are obtained as members of a family of commuting operators; their
diagonalization, however, remains an unsolved problem.
