  By mapping the Hamiltonians of the two-mode and 2-photon Rabi models to
differential operators in suitable Hilbert spaces of entire functions, we prove
that the two models possess entire and normalizable wavefunctions in the
Bargmann-Hilbert spaces only if the frequency $\omega$ and coupling strength
$g$ satisfy certain constraints. This is in sharp contrast to the quantum Rabi
model for which entire wavefunctions always exist. For model parameters
fulfilling the aforesaid constraints we determine transcendental equations
whose roots give the regular energy eigenvalues of the models. Furthermore, we
show that for $k\geq 3$ the $k$-photon Rabi model does not possess
wavefunctions which are elements of the Bargmann-Hilbert space for all
non-trivial model parameters. This implies that the $k\geq 3$ case is not
diagonalizable, unlike its RWA cousin, the $k$-photon Jaynes-Cummings model
which can be completely diagonalized for all $k$.
