Einstein–Weyl spaces and third-order differential equations

The three-dimensional null-surface formalism of Tanimoto [M. Tanimoto, “On the null surface formalism,” Report No. gr-qc/9703003 (1997)] and Forni et al. [Forni et al., “Null surfaces formation in 3D,” J. Math Phys. (submitted)] are extended to describe Einstein–Weyl spaces, following Cartan [E. Cartan, “Les espaces généralisées et l’integration de certaines classes d’equations différentielles,” C. R. Acad. Sci. 206, 1425–1429 (1938); “La geometria de las ecuaciones diferenciales de tercer order,” Rev. Mat. Hispano-Am. 4, 1–31 (1941)]. In the resulting formalism, Einstein–Weyl spaces are obtained from a particular class of third-order differential equations. Some examples of the construction which include some new Einstein–Weyl spaces are given.