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Mathematical ExpressionNecessary and Sufficient ConditionMeromorphic FunctionsBoundary Value ProblemBanaCh SpaceAlgebraic IndependenceSubject ClassificationDifferential EquationsS TheoremWeak SolutionMeromorphic FunctionExistence and UniquenessInitial DataExistence of Periodic SolutionsImaginary Quadratic FieldHamiltonian SystemsRiemannian ManifoldPeriodic SolutionExistence TheoremQualitative Analysis