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.