  Let $\mathcal{M}_{C}(2, 0)$ be the moduli space of semistable rank two and
degree zero Higgs bundles on a smooth complex hyperelliptic curve $C$ of genus
three. We prove that the quotient of $\mathcal{M}_{C}(2, 0)$ by a twisted
version of the hyperelliptic involution is an 18-dimensional holomorphic
symplectic variety admitting a crepant resolution, whose local model was
studied by Kaledin and Lehn to describe O'Grady's singularities. Similarly, by
considering the moduli space of Higgs bundles with trivial determinant
$\mathcal{M}_C(2, \mathcal{O}_{C})\subseteq \mathcal{M}_C(2, 0)$, we show that
the quotient of $\mathcal{M}_C(2, \mathcal{O}_{C})$ by the hyperelliptic
involution is a 12-dimensional holomorphic symplectic variety admitting a
crepant resolution.
