Abstract Let
. For
w ∈
K
1(3), a
-PBD is a pairwise balanced design on
v points with block size from the set
K
1(3) in which there is at least one block of size
w. In this paper, we investigate the existence problem for (
v,
K
1(3) ∪ {
w
*})-PBDs and give a complete solution to this problem. As its applications, we solve completely the embedding problem for directed
designs DB(4,1;
u)s. In addition, we also apply our (
v,
K
1(3) ∪ {
w
*})-PBDs to do embeddings for near resolvable triple systems and nested Steiner triple systems and give unified and simple
new proofs of two known theorems. Some new 4-GDDs are constructed as well.