We use N. Tanaka's theory of normal Cartan connections associated with simple graded Lie algebras to study Cartan's equivalence problem of single third order ordinary differential equations under contact transformations. As a result we obtain the complete structure equation with two differential inv riants, which is applied on general Legendre Grassmann bundles of three-dimensional contact manifolds.
Hajime SATO. Atsuko Yamada YOSHIKAWA. "Third order ordinary differential equations and Legendre connections." J. Math. Soc. Japan 50 (4) 993 - 1013, October, 1998. https://doi.org/10.2969/jmsj/05040993