The 3-way flower intersection problem for Steiner triple systems

The flower at a point x in a Steiner triple system (X, B) is the set of all triples containing x. Denote by J(F)(3)(r) the set of all integers k such that there exists a collection of three STS (2r + 1) mutually intersecting in the same set of k + r triples, r of them being the triples of a common flower. In this article we determine the set J(F)(3)(r) for any positive integer r equivalent to 0, 1 (mod 3) (only some cases are left undecided for r = 6, 7, 9, 24), and establish that J(F)(3)(r) = I-F(3)(r) for r equivalent to 0, 1 (mod 3) where I-F(3)(r) = {0, 1,..., 2r(r-1)/3 - 8, 2r(r-1)/3 - 6, 2r(r-1)/3}.