This paper is devoted to the study of rooted properties of phase surfaces defined by complex analytic systems. We first obtain the Rooted Theorem of Analytic Systems. Then we prove the Generalized Strong Rooted Theorem of (£") (m S 2), which implying the Strong Rooted Theorem of a Class of (£*). 1991 Mathematics subject classification: 34A20, 34C35, 58F25. Rooted theorem of analytic systems Consider differential equations in complex domain where C m denotes the m-dimensional complex spaces, and

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... F: C m -*C m is an analytic function. By Cauchy's theorem, equation ( 1.1) has a unique local analytic solution satisfying T = 0, W=W 0 , denoting W=W(T,W 0 ), \T\0). Taking W(T, W o ) as an element of analytic function, we obtain a global analytic function {T, W o ) with D(W 0 ) as its domain of definition. It can be proved that D(W 0 ) is an open subset of C 1 . If {T, W O )=W O , TeC\ and hence Proposition 1. Let I. be a phase surface of {I.I). If W o e2, then 2 Proof. For the simplicity, let Z 0 = E(W 0 ) and D 0 = D(W 0 ). For any W,eS 0 , there •Research supported by the National Science Foundation of P.R. China. 255