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For an arbitrary automorphism $\\theta$ of $G$, we introduce a holomorphic Poisson structure $\\pi_\\theta$ on $G$ which is invariant under the $\\theta$-twisted conjugation by $T$ and has the property that every $\\theta$-twisted conjugacy class of $G$ is a Poisson subvariety with respect to $\\pi_\\theta$. We describe the $T$-orbits of symplectic leaves, called $T$-leaves, of $\\pi_\\theta$ and compute the dimensions of the symplectic leaves (i.e, the ranks) of $\\pi_\\theta$. 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For an arbitrary automorphism $\\theta$ of $G$, we introduce a holomorphic Poisson structure $\\pi_\\theta$ on $G$ which is invariant under the $\\theta$-twisted conjugation by $T$ and has the property that every $\\theta$-twisted conjugacy class of $G$ is a Poisson subvariety with respect to $\\pi_\\theta$. We describe the $T$-orbits of symplectic leaves, called $T$-leaves, of $\\pi_\\theta$ and compute the dimensions of the symplectic leaves (i.e, the ranks) of $\\pi_\\theta$. 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