  We study two models describing the interaction of a two-level system with two
quantum field modes. The first one is equivalent to a dissipative two-state
system with just two boson fields in the absence of tunneling. The second
describes two orthogonal fields interacting with the corresponding orthogonal
dipoles of a two-level system. We show that both models present a partial
two-mode $SU(2)$ symmetry and that they can be solved in the exceptional case
of resonant fields. We study their ground state configurations, that is, we
find the quantum precursors of the corresponding semi-classical phase
transitions, as well as their whole spectra to infer their integrability. We
show that the first model in the exceptional case is isomorphic with the
quantum Rabi model and allows just two ground state configurations, vacuum and
non-vacuum. The second model allows four ground state configurations, one
vacuum, two non-vacuum single mode and one non-vacuum dual mode, and give
analytic and numerical pointers that may suggest its integrability. We also
show that in the single excitation subspace these models can serve as a fast
$SU(2)$ beam splitter even in the ultra-strong coupling regime.
