The metamorphosis of λ-fold block designs with block size four into λ-fold triple systems

Let (X,B) be a λ-fold block design with block size 4. If a star is removed from each block of B the resulting collection of triangles P(B) is a partial λ-fold triple system (X,P(B)). If the edges belonging to the deleted stars can be arranged into a collection of triangles T(B), then (X,P(B)∪T(B)) is a λ-fold triple system, called a metamorphosis of the λ-fold block design (X,B). The object of this paper is the complete solution of the following problem. For which n and λ does there exist a λ-fold block design with block size 4 having a metamorphosis into a λ-fold triple system?