  We consider solutions of the 2x2 matrix Hamiltonians of the physical systems
within the context of the su(2) and su(1,1) Lie algebra. Our technique is
relatively simple when compared with the others and treats those Hamiltonians
which can be treated in a unified framework of the $% Sp(4,R)$ algebra. The
systematic study presented here reproduces a number of earlier results in a
natural way as well as leads to a novel findings. Possible generalizations of
the method are also suggested.
