  We consider a finitely generated virtually abelian group $G$ acting properly
and without inversions on a CAT(0) cube complex $X$. We prove that $G$
stabilizes a finite dimensional CAT(0) subcomplex $Y \subseteq X$ that is
isometrically embedded in the combinatorial metric. Moreover, we show that $Y$
is a product of finitely many quasilines. The result represents a higher
dimensional generalization of Haglund's axis theorem.
