Analytical solution for the Davydov-Chaban Hamiltonian with sextic potential for $\gamma=30^{\circ}$

An analytical solution for the Davydov-Chaban Hamiltonian with a sextic oscillator potential for the variable $\beta$ and $\gamma$ fixed to $30^{\circ}$, is proposed. The model is conventionally called Z(4)-Sextic. For the considered potential shapes the solution is exact for the ground and $\beta$ bands, while for the $\gamma$ band an approximation is adopted. Due to the scaling property of the problem the energy and $B(E2)$ transition ratios depend on a single parameter apart from an integer number which limits the number of allowed states. For certain constraints imposed on the free parameter, which lead to simpler special potentials, the energy and $B(E2)$ transition ratios are parameter independent. The energy spectra of the ground and first $\beta$ and $\gamma$ bands as well as the corresponding $B(E2)$ transitions, determined with Z(4)-Sextic, are studied as function of the free parameter and presented in detail for the special cases. Numerical applications are done for the $^{128,130,132}$Xe and $^{192,194,196}$Pt isotopes, revealing a qualitative agreement with experiment and a phase transition in Xe isotopes.