The flower intersection problem for Kirkman triple systems

The flower at a point x in a Steiner triple system is the set of all triples containing x. Denote by I_R^*r] the set of all integers k such that there exists a pair of KTS(2r+1) having k+r triples in common, r of them being the triples of a common flower. In this article we determine the set I_R^*r] for any positive integer r≡1 (mod 3) (only nine cases are left undecided for r=7,13,16,19), and establish that I_R^*r]=Jr] for r≡1 (mod 3) and r22 where Jr]={0,1,…,2r(r?1)/3?6,2r(r?1)/3?4,2r(r?1)/3}.