  O'Grady showed that certain special sextics in $\mathbb{P}^5$ called EPW
sextics admit smooth double covers with a holomorphic symplectic structure. We
propose another perspective on these symplectic manifolds, by showing that they
can be constructed from the Hilbert schemes of conics on Fano fourfolds of
degree ten. As applications, we construct families of Lagrangian surfaces in
these symplectic fourfolds, and related integrable systems whose fibers are
intermediate Jacobians.
