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In this paper, we prove that $B_n$ admits no $\\mbox{Aut}(B_n)$-invariant strongly pseudoconvex complex Finsler metric other than a constant multiple of the Poincar$\\acute{\\mbox{e}}$-Bergman metric, while $P_n$ admits infinite many $\\mbox{Aut}(P_n)$-invariant complete strongly convex complex Finsler metrics other than the Bergman metric. 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Denote $\\mbox{Aut}(B_n)$ and $\\mbox{Aut}(P_n)$ the holomorphic automorphism group of $B_n$ and $P_n$ respectively. In this paper, we prove that $B_n$ admits no $\\mbox{Aut}(B_n)$-invariant strongly pseudoconvex complex Finsler metric other than a constant multiple of the Poincar$\\acute{\\mbox{e}}$-Bergman metric, while $P_n$ admits infinite many $\\mbox{Aut}(P_n)$-invariant complete strongly convex complex Finsler metrics other than the Bergman metric. 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