  We consider a generalization of the quantum Rabi model where the two-level
system and the single-mode cavity oscillator are coupled by an additional
Stark-like term. By adapting a method recently introduced by Braak [Phys. Rev.
Lett. {\bf 107}, 100401 (2011)], we solve the model exactly. The low-lying
spectrum in the experimentally relevant ultrastrong and deep strong regimes of
the Rabi coupling is found to exhibit two striking features absent from the
original quantum Rabi model: avoided level crossings for states of the same
parity and an anomalously rapid onset of two-fold near-degenerate levels as the
Rabi coupling increases.
