  In this work, the Davydov-Chaban Hamiltonian, describing the collective
motion of $\gamma$-rigid atomic nuclei, is amended by allowing the mass
parameter to depend on the nuclear deformation. Further, Z(4)-DDM
(Deformation-Dependent Mass) model is proposed by considering the Kratzer
potential for the $\beta$ variable, and solving the problem by techniques of
asymptotic iteration method (AIM). The results of the calculated spectra and
$B(E2)$ transition rates for series of $^{192-196}$Pt isotopes are compared
with the corresponding experimental data as well as with other theoretical
models. Exact analytical expressions are derived for spectra and normalized
wave functions of the Kratzer potential. The obtained results show an overall
good agreement with the experimental data and an important improvement in
respect to other models
