  We have constructed the quasi-exactly-solvable two-mode bosonic realizations
of su(2) and su(1, 1) algebra. We derive the relations leading to the
conditions for quasi-exact solvability of two-boson Hamiltonians by determining
a general procedure which maps the Schwinger representations of the su(2) and
su(1, 1) algebras to the Gelfand-Dyson representations, respectively. This
mapping allows us to study nonlinear quantum-optical systems in the framework
of quasi-exact solvability. Our approach also leads to a simple construction of
special functions of two variables which are the most appropriate functions to
study quasi-probabilities in quantum optics.
