TITLE: Infinitesimally Desarguian curve families
QUESTION [9 upvotes]: 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.

REPLY [2 votes]: A proof can be extracted from Cartan's "On manifolds with projective connections." (Translated by D. H. Delphenich).
In section VII Cartan defines a projective connection in the projectivised tangent bundle ("manifold of elements") with connection one-form:
$$
\omega = \begin{pmatrix}
\omega_0^0&\omega^1&\omega^2\\
\omega_1^0&\omega_1^1&\omega_1^2\\
\omega_2^0&\omega_2^1&\omega_2^2\\
\end{pmatrix}
$$
After normalising the connection he arrives at the curvature two-form
$$
\Omega = \begin{pmatrix}
\gamma&0&0\\
\Omega_1^0&\gamma&0\\
\Omega_2^0&\Omega_2^1&\gamma\\
\end{pmatrix},
$$
where
$$
\begin{aligned}
\Omega_2^1 &= a\omega^2\wedge\omega_1^2,\qquad a=-\frac{1}{6}\frac{\partial^4f}{\partial y'^4}\\
\Omega_1^0&=b\omega^1\wedge\omega^2,\qquad b=\text{complicated function}\\
\Omega_2^0&=\text{complicated two-form}\\
\gamma&=\Omega_0^0=\Omega_1^1=\Omega_2^2=\text{complicated two-form}.\\
\end{aligned}
$$
In section 23 Cartan then states the dual connection with connection one-form
$$
\varpi = \begin{pmatrix}
\varpi_0^0&\varpi^1&\varpi^2\\
\varpi_1^0&\varpi_1^1&\varpi_1^2\\
\varpi_2^0&\varpi_2^1&\varpi_2^2\\
\end{pmatrix}=\begin{pmatrix}
\omega_2^2&\omega_2^1&\omega^2\\
\omega_2^1&\omega_1^1&\omega^1\\
\omega_2^0&\omega_1^0&\omega_0^0\\
\end{pmatrix}
$$
which is simply the primal connection transposed with respect to the antidiagonal and the same goes for the the curvature two-form:
$$
\Pi = \begin{pmatrix}
\gamma&0&0\\
\Pi_1^0&\gamma&0\\
\Pi_2^0&\Pi_2^1&\gamma\\
\end{pmatrix} = \begin{pmatrix}
\gamma&0&0\\
\Omega_2^1&\gamma&0\\
\Omega_2^0&\Omega_0^1&\gamma\\
\end{pmatrix},
$$
so that we have
$$
\begin{aligned}
\Pi_1^0=\Omega_2^1\\
\Pi_2^1=\Omega_1^0
\end{aligned}
$$
We have that $\Omega_2^1=0$ is the condition that the right-hand side of the differential equation is a polynomial in $y'$ with degree at most 3 and the condition $\Omega_1^0=0$ is then "infinitesimal desargueness". In the dual connection these two conditions are swapped.
It might be argued that "infinitesimal nondesargueness" should really mean that any of $\Omega_2^1$ or $\Omega_1^0$ is non-vanishing since they are both obstructions to the construction of homogenous coordinates which then gives a projective structure on the $x,y$-plane with the solution curves as lines. The availability of homogenous coordinates is equivalent to the Desarguesian property of a projective plane.
Here's a couple of blog posts discussing these things:

The geometry of y''+y=0
The geometry of a second order differential equation, part 1
The geometry of a second order differential equation, part 2