  Let $E$ be a set of solids (hyperplanes) in $PG(4,q)$, $q$ even, $q>2$, such
that every point of $PG(4,q)$ lies in either $0$, $\frac12q^3$ or
$\frac12(q^3-q^2)$ solids of $E$, and every plane of $PG(4,q)$ lies in either
$0$, $\frac12q$ or $q$ solids of $E$. This article shows that $E$ is either the
set of solids that are disjoint from a hyperoval, or the set of solids that
meet a non-singular quadric $Q(4,q)$ in an elliptic quadric.
