A while ago I asked how to construct an infinite family of $(v,b,r,k,\lambda)$-designs satisfying $r=\lambda^{2}$ and got very good answers from Yuichiro Fujiwara and Ken W. Smith.

Now I'd like to up the ante and to generalize the question to $(r,\lambda)$-designs. Formally, we are talking here about a family $D$ of subsets of $\{1,2,\ldots,v\}$ such that:

(a) Each $i \in \{1,2,\ldots,v\}$ belongs to $r$ sets in $D$.

(b) Every two distinct $i,j \in \{1,2,\ldots,v\}$ belong together to $\lambda$ sets in $D$.

I am trying to find examples with $r=\lambda^{2}$, which is more difficult than in the case of $(v,k,\lambda)$ designs because this is more off the beaten path, so even stand-alone examples will be greatly appreciated.

P.S. As in the previous question, I require at least one pair of disjoint blocks, so most constructions based on symmetric designs will not apply here.