Comparison of Poisson structures on moduli spaces

Let $X$ be a complex irreducible smooth projective curve, and let ${\mathbb L}$ be an algebraic line bundle on $X$ with a nonzero section $\sigma_0$. Let $\mathcal{M}$ denote the moduli space of stable Hitchin pairs $(E,\, \theta)$, where $E$ is an algebraic vector bundle on $X$ of fixed rank $r$ and degree $\delta$, and $\theta\, \in\, H^0(X,\, End(E)\otimes K_X\otimes{\mathbb L})$. Associating to every stable Hitchin pair its spectral data, an isomorphism of $\mathcal{M}$ with a moduli space $\mathcal{P}$ of stable sheaves of pure dimension one on the total space of $K_X\otimes{\mathbb L}$ is obtained. Both the moduli spaces $\mathcal{P}$ and $\mathcal{M}$ are equipped with algebraic Poisson structures, which are constructed using $\sigma_0$. Here we prove that the above isomorphism between $\mathcal{P}$ and $\mathcal{M}$ preserves the Poisson structures.