  We show that if $r\geq 3$ and $\alpha$ is a faithful $Z^r$-Cartan action on a
torus $T^d$ by automorphisms, then any closed subset of $(T^d)^2$ which is
invariant and topologically transitive under the diagonal $\bZ^r$-action by
$\alpha$ is homogeneous, in the sense that it is either the full torus
$(T^d)^2$, or a finite set of rational points, or a finite disjoint union of
parallel translates of some d-dimensional invariant subtorus. A counterexample
is constructed for the rank 2 case.
