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 {\it 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 {\it dual} curve family.
It has been stated without proof for instance in Gelfand, Gindikin, and Shapiro, {\it A local problem of integral geometry in a space of curves}, Functional Anal.\ Appl.\ {\bf 13} (1979), p.\ 88 and p.\ 99, and in the book by Arnold, {\it Geometrical methods in the theory of ordinary differential equations}, ch.\ 1, $\S$ 6 G Exercise (!), 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.

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, mathnet) and in the book by Arnold, Geometrical methods in the theory of ordinary differential equations, ch. 1, § 6 G Exercise (!), 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.