Packing Integral Tori in Del Pezzo Surfaces

We extend a packing result of R. Hind and E. Kerman for integral Lagrangian tori in ${\mathbb{S}}^2\times {\mathbb{S}}^2$ to the Del Pezzo surfaces $({\mathbb{D}}_n,{\omega }_{{\mathbb{D}}_n})$ for $n=1,\dots ,5$. An integral torus is one whose relative area homomorphism is integer-valued, and we seek a maximal integral packing. By definition, this is a disjoint collection $\{L_i\}$ of integral Lagrangian tori with the following property: any other integral Lagrangian torus not in this collection must intersect at least one of the $L_i$. We show that one can always find such a packing consisting of only the Clifford torus.