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In this paper we prove that the set of $S$-integral points of the smoothcubic surfaces in $\mathbb{A}^3$ over a number field $k$ is not thin, forsuitable $k$ and $S$. As a corollary, we obtain results on the complement in$\mathbb{P}^2$ of a smooth cubic curve, improving on Beukers' proof that the$S$-integral points are Zariski dense, for suitable $S$ and $k$. With ourmethod we reprove Zariski density, but our result is more powerful since it isa stronger form of Zariski density. We moreover prove that the rational integerpoints on the Fermat cubic surface $x^3+y^3+z^3=1$ form a non-thin set and welink our methods to previous results of Lehmer, Miller-Woollett and Mordell.

The Hilbert Property for integral points of affine smooth cubic surfaces