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