On the geometry of quadratic second-order Abel ordinary differential equations

In this paper, we study the contact geometry of second-order ordinary differential equations that are quadratic in the highest derivative (the so-called quadratic Abel equations). Namely, we realize each quadratic Abel equation as the kernel of some nonlinear differential operator. This operator is defined by a quadratic form on the Cartan distribution in the $1$-jet space. This observation makes it possible to establish a one-to-one correspondence between quadratic Abel equations and quadratic forms on Cartan distribution. Using this realization, we construct a contact-invariant $\{e\}$-structure associated with a nondegenerate Abel equation (i.e., basis of vector fields that is invariant under contact transformations). Finally, in terms of this $\{e\}$-structure we solve the problem of contact equivalence of nondegenerate Abel equations.