  Let $p$ be an odd prime. In this paper we provide a construction which gives
four non-Schurian association schemes for every $p\geq 5$ and two for $p=3$.
This construction is explained using incidences between points and lines of a
biaffine plane and we also provide a pure algebraic model for it with the aid
of finite Heisenberg groups. The obtained results are discussed in a more wide
framework.
