  With employing the SO(6)representation of eigenstates, the energy spectra and
quadrupole transition rates of SU(3)<->SU(3)* transition region are considered
for systems with total boson number N=2,3,4. The apparent level crossing and
significant variations in the transition rates for X=0, namely, the critical
point of transitional Hamiltonian, suggest a phase transition between these
limits. The parameter free (up to overall scale factors) predictions for
spectra and B(E2)rates by X=0are found to be in good agreement with those for
nuclei provide empirical evidences for O(6)dynamical symmetry, namely
Hg(200-204)isotopes.
