  Generators of the quantum $SU_q(2)$ algebra are obtained in the explicit form
in the basis where the operator $\exp {J_z\over 2} J_x \exp {J_z\over 2}$ is
diagonal. It is shown that the solution of this problem is related to the
representation theory of the two-dimensional algebra $[s,r]=\tanh t
(s^2-r^2+1)$. The relevance of such basis to some problems of quantum optics is
discussed.
