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They gave the possibilities of the birational models $W$ of the quotients and the branch divisors $B_0$ induced by involutions $\\sigma$ on the surfaces $S$.\n  In this paper we improve and refine the results of Lee and the second named author. We exclude the case of the Kodaira dimension $\\kappa(W)=1$ when the number $k$ of isolated fixed points of an involution $\\sigma$ on $S$ is nine. 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