Exact isolated solutions for the two-photon Rabi Hamiltonian
  The two-photon Rabi Hamiltonian is a simple model describing the interaction of light with matter, with the interaction being mediated by the exchange of two photons. Although this model is exactly soluble in the rotating-wave approximation, we work with the full Hamiltonian, maintaining the non-integrability of the model. We demonstrate that, despite this non-integrability, there exist isolated, exact solutions for this model analogous to the so-called Juddian solutions found for the single-photon Rabi Hamiltonian. In so doing we use a Bogoliubov transformation of the field mode, as described by the present authors in an earlier publication. 
C. Emary and R. F. Bishop