  For integers $k$, we consider the affine cubic surface $V_{k}$ given by
$M({\bf x})=x_{1}^2 + x_{2}^2 +x_{3}^2 -x_{1}x_{2}x_{3}=k$. We show that for
almost all $k$ the Hasse Principle holds, namely that $V_{k}(\mathbb{Z})$ is
non-empty if $V_{k}(\mathbb{Z}_p)$ is non-empty for all primes $p$, and that
there are infinitely many $k$'s for which it fails. The Markoff morphisms act
on $V_{k}(\mathbb{Z})$ 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.
