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

Elekes-Szab\'o for collinearity on cubic surfaces