  We construct infinitely many Einstein-Weyl structures on $S^2 \times R$ of
signature (-++) which is sufficiently close to the model case of constant
curvature, and whose space-like geodesics are all closed. Such structures are
obtained from small perturbations of the diagonal of $CP^1 \times CP^1$ using
the method of LeBrun-Mason type twistor theory. The geometry of constructed
Einstein-Weyl space is well understood from the configuration of holomorphic
disks. We also review Einstein-Weyl structures and their properties in the
former half of this article.
