A (2, 3)-packing on X is a pair (X, A), where A is a set of 3-subsets (called blocks) of X, such that any pair of distinct points from X occurs together in at most one block. Its leave is a graph (X, E) such that E consists of all the pairs which do not appear in any block of A. In this article, we shall construct a set of 6k − 2 disjoint (2, 3)-packings of order 6k + 4 with K 1,3 ∪ 3kK 2 or G 1 ∪ (3k − 1)K 2 as their common leave for any integer k ≥ 1 with a few possible exceptions (G 1 is a special graph of order 6). Such a system can be used to construct perfect threshold schemes as noted by Schellenberg and Stinson (1989).