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We prove several results concerning the Hitchin map on $T^*\\!\\mathrm{Bun}_{\\mathcal{G}}$. We first show that the parahoric analogue of the global nilpotent cone is isotropic and use this to prove that $\\mathrm{Bun}_{\\mathcal{G}}$ is \"very good\" in the sense of Beilinson-Drinfeld. We then prove that the parahoric Hitchin map is a Poisson map whose generic fibres are abelian varieties. 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