  Starting from integrable $su(2)$ (quasi-)spin Richardson-Gaudin XXZ models we
derive several properties of integrable spin models coupled to a bosonic mode.
We focus on the Dicke-Jaynes-Cummings-Gaudin models and the two-channel
$(p+ip)$-wave pairing Hamiltonian. The pseudo-deformation of the underlying
$su(2)$ algebra is here introduced as a way to obtain these models in the
contraction limit of different Richardson-Gaudin models. This allows for the
construction of the full set of conserved charges, the Bethe Ansatz state, and
the resulting Richardson-Gaudin equations. For these models an alternative and
simpler set of quadratic equations can be found in terms of the eigenvalues of
the conserved charges. Furthermore, the recently proposed eigenvalue-based
determinant expressions for the overlaps and form factors of local operators
are extended to these models, linking the results previously presented for the
Dicke-Jaynes-Cummings-Gaudin models with the general results for
Richardson-Gaudin XXZ models.
