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We recently obtained generalized Marino-Prodi perturbation theorem which is employed to discover some relationships between Morse Indices of the solutions and topology properties of level set of energy functional corresponding to superlinear elliptic equation.To get the existence of infinitely many distinct solutions,people usually treat superlinear elliptic problems with symmetry by resorting to genus and index theory.It is natural ask if we can get the nontriviality of singular homology groups of level set of even functional.One of the purposes of our work is to explore topological behaviour of even functional and disclose the property.