  We study the orchard problem on cubic surfaces. We classify possibly
reducible cubic surfaces $X\subseteq \mathbb{P}^3(\C)$ with smooth components
on which there exist families of finite sets (of unbounded size) with
quadratically many 3-rich lines which do not concentrate (in a natural sense)
on any projective plane. Namely, we prove that such a family exists precisely
when $X$ is a union of three planes sharing a common line.
  Along the way, we obtain a general result about nilpotency of groups
admitting an algebraic action satisfying an Elekes-Szab\'o condition, and we
prove the following purely algebrogeometric statement: if the composition of
four Geiser involutions through sufficiently generic points $a,b,c,d$ on a
smooth irreducible cubic surface has infinitely many fixed points, then a
single plane contains $a,b,c,d$ and all but finitely many of the fixed points.
