A very special EPW sextic and two IHS fourfolds

We show that the Hilbert scheme of two points on the Vinberg K3 surface has a two-to-one map onto a very symmetric EPW sextic Y in P-5. The fourfold Y is singular along 60 planes, 20 of which form a complete family of incident planes. This solves a problem of Morin and O'Grady and establishes that 20 is the maximal cardinality of such a family of planes. Next, we show that this Hilbert scheme is birationally isomorphic to the Kummer-type IHS fourfold X-0 constructed by Donten-Bury and Wisniewski [On 81 symplectic resolutions of a 4-dimensional quotient by a group of order 32, preprint (2014)]. We find that X-0 is also related to the Debarre-Varley abelian fourfold.