Representation of the quantum algebra $SU_q(2)$ in the basis with diagonal $"J_x"$ generator

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.