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 and the theory of integrable systems. 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.

我们将Cayley结构定义为四维流形上Cayley直纹立方表面的一个域，并通过显示它们与不确定保形结构的相似性以及它们与微分方程和可积系统的联系来激发它们的研究。特别是，对于Cayley结构，引入了为维度4中的不确定保形结构定义的某些概念的扩展，例如，半平坦度，零叶面的存在，超半平坦度，相关的二阶ODE对和非分散Lax对。解决了等价问题后，我们获得了基本不变式，找到了几类Cayley结构的局部普遍性并给出了例子。