Packing Lagrangian tori

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.