  In this paper we consider the problem of packing a symplectic manifold with
integral Lagrangian tori, that is Lagrangian tori whose area homomorphsims take
only integer values. We prove that the Clifford torus in $S^2 \times S^2$ is a
maximal integral packing, in the sense that any other integral Lagranian torus
must intersect it. In the other direction, we show that in any symplectic
polydisk $P(a,b)$ with $a,b>2$, there is at least one integral Lagrangian torus
in the complement of the collection of standard product integral Lagrangian
tori.
