The metamorphosis of l-fold block designs with block size four into l-fold 4-cycle systems

Let (X, B) be a lambda-fold block design with block size 4. If a path of length two is removed from each block of B the resulting collection of kites K is a partial lambda-fold kite system (X, K). If the deleted edges can be arranged into a collection of kites D, then (X, K boolean OR D) is a lambda-fold kite system [5]. Now for each block b is an element of B let {P-1(b), P-2(b), P-3(b)} be a partition of b into paths of length two and define for each i =1, 2, 3, sets K-i and D-i as follows: for each b is an element of B, put the kite b\P-i(b) in K-i and the two edges belonging to the path P-i(b) in D-i. If the edges in D-i can be arranged into a collection of kites D-i* then M-i = (X, K-i boolean OR D-i*) is a lambda-fold kite system, called the ith metamorphosis of (X, B). The full metamorphosis is the set of three metamorphoses {M-1, M-2, M-3}. We give a complete solution of the following problem: for which n and lambda does there exist a lambda-fold block design with block size 4 having a full metamorphosis into a lambda-fold kite system