  A method is developed to determine the eigenvalues and eigenfunction of
two-boson $2\times 2$ matrix Hamiltonians include a wide class of quantum
optical models. The quantum Hamiltonians have been transformed in the form of
the one variable differential equation and the conditions for its solvability
have been discussed. We present two different transformation procedure and we
show our approach unify various approaches based on Lie algebraic technique. As
an application, solutions of the modified Jaynes-Cummings and two-level Jahn
Teller Hamiltonians are studied.
