Einstein-Weyl spaces and (1, n) -curves in the quadric surface

Following the ideas of Hitchin on the twistoral approach to 3-dimensional Einstein-Weyl geometry we construct a series of complex surfaces containing rational curves with self-intersection number 2. These mini twistor spaces are obtained by taking an n-fold covering of a neighbourhood of a (1, n)- curve in the quadric CP₁ x CP₁ branched along the curve. We describe the corresponding Einstein-Weyl geometry on the parameter space of curves.