Nonabelianization of Higgs bundles

The character varieties of representations of a surface group into the Lie groups SL(m,H), SO(2m,H) and Sp(m,m) have a holomorphic description in terms of the moduli space of Higgs bundles. We show that the fibres of the integrable system in these cases are not abelian varieties, but are instead moduli spaces of rank 2 bundles on a spectral curve, satisfying natural stability conditions.