We consider the integrable system on the moduli space of Higgs bundles restricted to the subvariety corresponding to representations of a surface group into certain noncompact real forms, and in doing so encounter as the fiber moduli spaces of rank $2$ bundles on a spectral curve. This contrasts with the general case where the fiber is an abelian variety.
"Nonabelianization of Higgs bundles." J. Differential Geom. 97 (1) 79 - 89, May 2014. https://doi.org/10.4310/jdg/1404912103