  Based on the boson realization of the Euclidean algebras, it is found that
the E($n$) dynamical symmetry (DS) may emerge at the critical point of the
U($n$)-SO($n+1$) quantum phase transition. To justify this finding, we provide
a detailed analysis of the critical dynamics in the U($n+1$) vibron model in
both quantal and classical ways. It is further shown that the low-lying
structure of $^{82}$Kr may serve as an excellent empirical realization of the
E(5) DS in experiments.
