  We study the properties of the low-lying states at the critical point of the
phase transition from U(3) to O(4) symmetry in the U(4) vibron model in detail.
By analyzing the general characteristics and comparing the calculated results
of the energy spectra and the E1, E2 transition rates in E(3) symmetry, in
$r^4$ potential model and the finite boson number case in boson space, we find
that the results in the $r^4$ potential demonstrates the characteristic of the
classical limit at the critical point well and the E(3) symmetry over-predict
the energy levels and under-predict the E1 and E2 transition rates of the
states at the critical point. However, the E(3) symmetry may describe part of
the properties of the system with boson number around 10 to 20. We also confirm
that the 12C+12C system is an empirical evidence of the state at the critical
point of the phase transition in the U(4) model when concerning the energies of
the low-lying resonant states.
