The (driven) Rabi model, together with its two-mode, two-photon, and asymmetric generalizations, are exotic examples of quasi-exactly solvable models in that a corresponding 2nd order ordinary differential equation (ODE) ${\cal L}\psi=0$ with polynomial coefficients (i) is not Fuchsian one and (ii) the differential operator ${\cal L}$ comprises energy E dependent terms $\sim Ez d_z$, $Ez$, $E^2$. When recast into a Schr\"odinger equation (SE) form with the first derivative term being eliminated and the coefficient of $d_x^2$ set to one, such an equation is characterized by a nontrivially energy dependent potential. The concept of a gradation slicing is introduced to analyze polynomial solutions of such equations. It is shown that the ODE of all the above Rabi models are characterized by the same unique set of grading parameters. General necessary and sufficient conditions for the existence of a polynomial solution are formulated. Unlike standard eigenvalue problems, the condition that 查看全文>>