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

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