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Based on the boson realization of the Euclidean algebras, it is found that the E(<i>n</i>) dynamical symmetry (DS) may emerge at the critical point of the U(<i>n</i>)-SO(<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></semantics></math></inline-formula>) quantum phase transition. To justify this finding, we provide a detailed analysis of the transitional Hamiltonian in the U(<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></semantics></math></inline-formula>) vibron model in both quantal and classical ways. It is further shown that the low-lying structure of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow></mrow><mn>82</mn></msup></semantics></math></inline-formula>Kr can serve as an excellent empirical realization of the E(5) DS, which provides a specific example of the Euclidean DS in experiments.