  Spectral properties of nuclei near the critical point of the quantum phase
transition between spherical and axially symmetric shapes are studied in a
hybrid collective model which combines the $\gamma$-stable and $\gamma$-rigid
collective conditions through a rigidity parameter. The model in the lower and
upper limits of the rigidity parameter recovers the X(5) and X(3) solutions
respectively, while in the equally mixed case it corresponds to the X(4)
critical point symmetry. Numerical applications of the model on nuclei from
regions known for critical behavior reveal a sizable shape phase mixing and its
evolution with neutron or proton numbers. The model also enables a better
description of energy spectra and electromagnetic transitions for these nuclei.
