On the Integrability of the Rabi Model

The exact spectrum of the Rabi hamiltonian is analytically found for arbitrary coupling strength and detuning. I present a criterion for integrability of quantum systems containing discrete degrees of freedom which shows that in this case a finite symmetry group may entail integrability, even without the presence of conserved charges beyond the hamiltonian itself. Moreover, I introduce and solve a natural generalization of the Rabi model which has no symmetries and is therefore probably the smallest non-integrable physical system.