  We construct five families of two-dimensional moduli spaces of parabolic
Higgs bundles (respectively local systems) by taking the equivariant Hilbert
scheme of a certain finite group acting on the cotangent bundle of an elliptic
curve. We show that the Hilbert scheme of m points of these surfaces is again a
moduli space of parabolic Higgs bundles (respectively local systems),
confirming a conjecture of Boalch in these cases and extending a result of
Gorsky--Nekrasov--Rubtsov. Using the McKay correspondence we establish the
autoduality conjecture for the derived categories of the moduli spaces of Higgs
bundles under consideration.
