About a decomposition of the group ZpqZpq, where pp and qq are distinct odd primes

In this paper we are interested in finding a partition of ZnZn such that the component sets satisfy two properties: the first one says that the double of each element of every set is congruent modulo nn with the consecutive element of the same set; the second property states that there exist two distinct elements in each set such that their sum modulo nn belongs to the successive set. What is this decomposition useful for? It permits one to find the structure of a complete hypergroup HH in some particular cases.