For integers k, we consider the affine cubic surface Vk given by M(x)=x12+x22+x32-x1x2x3=k. We show that for almost all k the Hasse Principle holds, namely that Vk(Z) is non-empty if Vk(Zp) is non-empty for all primes p, and that there are infinitely many k’s for which it fails. The Markoff morphisms act on Vk(Z) with finitely many orbits and a numerical study points to some basic conjectures about these “class numbers” and Hasse failures. Some of the analysis may be extended to less special affine cubic surfaces.
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