The $2$-dimensional family of solution curves $y = u(x, \xi, \eta) \approx \eta + \xi x + \mathcal O(x^2)$ near $y = 0$ of a differential equation   
$y'' = \Phi(x, y, y')$
has been called *infinitesimally Desarguian* if 
\begin{equation*}
 \Phi(x, y, p) = \mathcal O(|y|^3 + |p|^3) \quad \textrm{as $(y, p) \to (0, 0)$} .  
\end{equation*}
The reason for this terminology is that such curve families in some sense are close to the family of straight lines. The curve family in the $\xi \eta$-plane that is defined by $y = u(x, \xi, \eta)$ while  $x$ and $y$ are considered as parameters is called the *dual* curve family.  
It has been stated without proof for instance in  Gelfand, Gindikin, and Shapiro, *A local problem of integral geometry in a space of curves*, Functional Anal. Appl. **13** (1979),  p.  88 and p.  99 ([doi: 10.1007/BF01077241](https://doi.org/10.1007/BF01077241), [mathnet](http://mi.mathnet.ru/faa1896)) and in the book by Arnold, *Geometrical methods in the theory of ordinary differential equations*, [ch. 1, § 6 G Exercise (!)](https://books.google.com/books?id=nejlBwAAQBAJ&pg=PA56), that a curve family is infinitesimally Desarguian if and only if its dual family is defined by a differential equation of the form 
$\eta'' = \Psi(\xi, \eta, \eta') $, 
where $\Psi(\xi, \eta, p)$ is a polynomial in $p$ of degree at most $3$ with coefficients that are smooth functions of $(\xi, \eta)$; that class is known to be preserved by arbitrary smooth diffeomorphisms in the $\xi \eta$ plane, see the cited book by Arnold.

I would very much like to know a reference for the proof of this fact.