  An infinite family of exact solutions of the two-photon Rabi model was found
by investigating the differential algebraic properties of the Hamiltonian. This
family corresponds to energy level crossings not covered by the Juddian class,
which is given by elemetary functions. In contrast, the new states are
expressible in terms of parabolic cylinder or Bessel functions. We discuss
three approaches for discovering this hidden structure: factorization of
differential equations, Kimura transformation, and a doubly-infinite,
transcendental basis of the Bargmann space.
