  In this paper we study a general Hamiltonian with a linear structure given in
terms of two different realizations of the $SU(1,1)$ group. We diagonalize this
Hamiltonian by using the similarity transformations of the $SU(1,1)$ and
$SU(2)$ displacement operators performed to the $su(1,1)$ Lie algebra
generators. Then, we compute the Berry phase of a general time-dependent
Hamiltonian with this general $SU(1,1)$ linear structure.
