Interesting geometric application of Hitchin Fibration

You may know this already, but I'm not sure if you'll get many other answers. A nice result of Brunebarbe, Klingler and Totaro [Symmetric differentials and the fundamental group, Duke 2013] is that if the fundamental group of a smooth complex projective variety $X$ has a finite dimensional representation with infinite image, then $X$ carries a nonzero holomorphic symmetric differential form, i.e. $H^0(X, sym^i\Omega_X^1)\not=0$ for some $i>0$. As I recall, one step of their proof uses a special case, due to me, that the same conclusion holds if $\pi_1(X)$ has a nonrigid representation. The latter assumption amounts to saying that the "Betti" moduli space $$M_{Betti}(X,n)=Hom(\pi_1(X),GL_n(\mathbb{C}))//GL_n(\mathbb{C})$$ is an infinite set. The proof of the special case is easy, modulo Simpson's work. Since $M_{Betti}(X,n)$ is affine, and infinite, it must be noncompact in the classical topology. But $M_{Betti}(X,n)^{an}$ is homeomorphic to $M_{Dol}(X,n)^{an}$ (the moduli space of rank $n$ Higgs bundles with the usual conditions). Since the Hitchin map $H_2$ is proper, the non compactness of $M_{Dol}(X,n)^{an}$ forces $H_2$ to have infinite image, so $sym^i\Omega_X$ must have sections for some $i$ in the range $i=1,\ldots, n$.

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.