TITLE: Interesting geometric application of Hitchin Fibration
QUESTION [10 upvotes]: Let $X$ be a smooth complex projective variety. Let $M_{Higgs}(X, P)$ be the coarse moduli which universally corepresents the functor:
$$M^{\#}_{Higgs}(X, P): Sch/\mathbb{C}\longrightarrow \mathcal{Set}$$
which assigns to any scheme $S$ the set of isomorphism classes of semi-stable Higgs bundles $(E, \theta)$ over $X\times S$ with Hilbert poly $P$. Here $P=nP_0$, with $P_0$ being the Hilbert polynomial of $\mathcal{O}_X$ with respect to a fixed polarization $\mathcal{O}_X(1)$.
Then we have the proper Hitchin map $$H_1: M_{Higgs}(X, P) \rightarrow \bigoplus_{i=1}^nH^0(X, sym^i\Omega^1(X))$$
which maps $(E,\theta)$ to the characteristic polynomial of $\theta$. (Take the coefficients.)
Consider the Dolbeault Moduli space $M_{Dol}(X, P)$, which universally corepresents $$M^{\#}_{Dol}(X, P): Sch/\mathbb{C}\longrightarrow \mathcal{Set}$$
which assigns to any scheme $S$ the set of isomorphism classes of semi-stable Higgs bundles $(E, \theta)$ over $X\times S$ with Hilbert polynomial $P$ for which the Chern classes of $E$ vanish. 
Since $M_{Dol}(X, P)$ is disjoint union of some of the connected components of $M_{Higgs}(X, P)$, we should also have the following proper Hitchin map
$$H_2: M_{Dol}(X, P) \rightarrow \bigoplus_{i=1}^nH^0(X, sym^i\Omega^1(X))$$
Can any one give some interesting, even small, geometric applications of the Hitchin map $H_1$ or $H_2$? Any answers and references are appreciated.

REPLY [2 votes]: If $X$ is a smooth projective curve with genus $g \geq 2$ and $M_{Higgs}(X,P)$ is smooth, the map:
$$ H_1 : M_{Higgs}(X,P) \longrightarrow \bigoplus_{i} H^0(X, Sym^i \Omega_X)$$ gives $M_{Higgs}(X,P)$ a structure of complete algebraic integrable system. I think this result is due to Beauville. See : https://projecteuclid.org/download/pdf_1/euclid.acta/1485890603. As far as I know, it is quite difficult to provide examples of complete algebraic integrable systems. 
EDIT : as Beauville comments below, the algebraic complete integrability of this moduli space was proved by Hitchin himself.