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Denote by $E(G,O)$ the space of germs of $C^{\\infty}$ diffeomorphisms $(\\mathbb{R}^2,O)\\to(\\mathbb{R}^2,O)$ that preserve orbits of $G$. Let also $E_{\\mathrm{id}}(G,O)$ be the identity component of $E(G,O)$ with respect to $C^1$-topology.\n  Suppose that $g$ has no multiple prime factors. 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