Triple metamorphosis of twofold triple systems

In a simple twofold triple system (X,B), any two distinct triples T1, T2 with |T1∩T2|=2 form a matched pair. Let F be a pairing of the triples of B into matched pairs (if possible). Let D be the collection of double edges belonging to the matched pairs in F, and let F∗ be the collection of 4-cycles obtained by removing the double edges from the matched pairs in F. If the edges belonging to D can be assembled into a collection of 4-cycles D∗, then (X,F∗∪D∗) is a twofold 4-cycle system called a metamorphosis of the twofold triple system (X,B). Previous work (Gionfriddo and Lindner, 2003  [7]) has shown that the spectrum for twofold triple systems having a metamorphosis into a twofold 4-cycle system is precisely the set of all n≡0,1,4 or 9(mod12), n≥9. In this paper, we extend this result as follows. We construct for each n≡0,1,4 or 9(mod12), n≠9 or 12, a twofold triple system (X,B) with the property that the triples in B can be arranged into three sets of matched pairs F1, F2, F3 having metamorphoses into twofold 4-cycle systems (X,F1∗∪D1∗), (X,F2∗∪D2∗), and (X,F3∗∪D3∗), respectively, with the property that D1∪D2∪D3=2Kn. In this case we say that (X,B) has a triple metamorphosis. Such a twofold triple system does not exist for n=9, and its existence for n=12 remains an open and apparently a very difficult problem.