  The various generalized spectral determinants (G-functions) of the two-photon
quantum Rabi model are analyzed with emphasis on the qualitative aspects of the
regular spectrum. Whereas all of them yield at least a subset of the exact
regular eigenvalues, only the G-function proposed by Chen et al. in 2012
exhibits an explicitly known pole structure which dictates the approach to the
collapse point. We derive this function rigorously employing the
$\mathbb{Z}_4$-symmetry of the model and show that its zeros correspond to the
complete regular spectrum.
