[en] An exactly solvable version of the Bohr Hamiltonian is proposed for a unified treatment of transitional nuclei. The model employs an energy-dependent potential in the $\mathrm{\beta <!--\; ??\; -->}$ shape variable which is amended with a centrifugal contribution from the $\mathrm{\gamma <!--\; ??\; -->}$ and angular degrees of freedom. The parametrization of the $\mathrm{\gamma <!--\; ??\; -->}$ part of the potential enables the connection between the adiabatic separation of $\mathrm{\beta <!--\; ??\; -->}$ and $\mathrm{\gamma <!--\; ??\; -->}$ fluctuations suitable for critical point nuclei and the exactly separable solutions specific to deformed nuclei. The fully analytical model is applied to known critical point nuclei as well as a neighbouring isotope for each case. Beside very good agreement with experiment for energies and electromagnetic transitions, numerical results also reveal specific model characteristics for critical nuclei in comparison to more or less deformed nuclei.