  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.
