The double EPW cube of a general Gushel-Mukai fourfold is the MRC quotient of the Hilbert scheme of twisted cubics, and it admits a Lagrangian covering family.
On the symplectic eightfold associated to a P faffian cubic fourfold
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Double EPW cubes from twisted cubics on Gushel-Mukai fourfolds
The double EPW cube of a general Gushel-Mukai fourfold is the MRC quotient of the Hilbert scheme of twisted cubics, and it admits a Lagrangian covering family.