  We show that if $f(u)\in \mathbb{Z}[u]$ is a monic cubic polynomial, then for
all but finitely many $n\in \mathbb{Z}$ the affine cubic surface
$f(u_{1})+f(u_{2})+f(u_{3})=n \subset \mathbb{A}^{3}_{\mathbb{Z}}$ has no
integral Brauer-Manin obstruction to the Hasse principle.
