The action of contact transformations pseudogroup on the second order   ODEs which are cubic in second derivative
  In the present paper we consider the problem of local equivalence of second order ODEs which are cubic in second derivative under the action of the pseudogroup of contact transformations. We show how it may be reduced to the equivalence problem of 2-webs in $\mathbb{R}^3$ under the action of finite-dimensional group, admitted by the associated second order equation. 
Vadim V. Shurygin Jr
math.DG