  Let ${\cal B}$ be a nontrivial biplane of order $k-2$ represented by
symmetric canonical incidence matrix with trace $1+ \binom{k}{2}$. We proved
that ${\cal B}$ includes a partially balanced incomplete design with
association scheme of three classes. Consequently, these structures are
symmetric, having $2k-6$ points. While it is not known whether this class is
finite or infinite, we show that there is a related superclass with infinitely
many representatives.
