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

In this paper we are interested in finding a partition of Z n such that the component sets satisfy two properties: the first one says that the double of each element of every set is congruent modulo n 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 n belongs to the successive set. What is this decomposition useful for? It permits one to find the structure of a complete hypergroup H in some particular cases.