  Given a smooth projective complex curve $X$ with an involution $\sigma$, we
study the Hitchin systems for the locus of anti-invariant (resp. invariant)
stable vector bundles over $X$ under $\sigma$. Using these integrable systems
and the theory of the nilpotent cone, we study the irreducibility of these
loci. The anti-invariant locus can be thought of as a generalisation of Prym
varieties to higher rank.
