  A path system $\mathcal{P}$ in a graph $G=(V,E)$ is said to be irreducible if
there does not exist a partition $V= A\sqcup B$ such that $\mathcal{P}$
restricts to a path system on both $G[A]$ and $G[B]$. In this paper, we
construct an infinite family of non-metrizable irreducible path systems defined
on certain Paley graphs.
