The Cayley cubic and differential equations

Understand

We define Cayley structures as a field of Cayley's ruled cubic surfaces over a four dimensional manifold and motivate their study by showing their similarity to indefinite conformal structures and their link to differential equations.

• In particular, for Cayley structures an extension of certain notions defined for indefinite conformal structures in dimension four are introduced, e.g., half-flatness, existence of a null foliation, ultra-half-flatness, an associated pair of second order ODEs, and a dispersionless Lax pair.
• After solving the equivalence problem we obtain the fundamental invariants, find the local generality of several classes of Cayley structures and give examples.