A complex surface which has a smooth rational curve whose self-intersection number is two is called a minitwistor space. These rational curves constitute a 3-dimensional family, and it is known that the parameter space of this family is naturally equipped with a special geometric structure, called an Einstein-Weyl structure (Hitchin 1982). About 10 years ago, in collabora- tion with F. Nakata, we showed that the same result as Hitchin holds even when we allow a correct number of ordinary double points to the rational curves in a complex surface. It seems natural to call the number of ordinary double points the genus of the minitwistor space. In this talk, we present our recent result that minitwistor spaces with genus one are exactly quartic surfaces in CP4 which are called Segre surfaces, and the Einstein-Weyl spaces corresponding to Segre surfaces are Zariski open subsets of the dual varieties of the Segre surfaces. Details can be found in arXiv:2009.05242.