On contact equivalence of Abel cubic differential equations of the second order

We study the geometry of cubic Abel differential equations on one-dimensional real curve. We prove that such an equation is the kernel of some non-linear differential operator. This operator is defined by a cubic on Cartan distribution in $1$-jet space. With the help of this observation we construct contact-invariant $\{e\}$-structure associated with non-degenerated Abel equation and obtain contact classification of such equations.