  In the geometry of generic 2-plane fields on 5-manifolds, the local
equivalence problem was solved by Cartan who also constructed the fundamental
curvature invariant. For generic 2-plane fields or $(2,3,5)$-distributions
determined by a single function of the form $F(q)$, the vanishing condition for
the curvature invariant is given by a 6$^{\rm th}$ order nonlinear ODE.
Furthermore, An and Nurowski showed that this ODE is the Legendre transform of
the 7$^{\rm th}$ order nonlinear ODE described in Dunajski and Sokolov. We show
that the 6$^{\rm th}$ order ODE can be reduced to a 3$^{\rm rd}$ order
nonlinear ODE that is a generalised Chazy equation. The 7$^{\rm th}$ order ODE
can similarly be reduced to another generalised Chazy equation, which has its
Chazy parameter given by the reciprocal of the former. As a consequence of
solving the related generalised Chazy equations, we obtain additional examples
of flat $(2,3,5)$-distributions not of the form $F(q)=q^m$. We also give
4-dimensional split signature metrics where their twistor distributions via the
An-Nurowski construction have split $G_2$ as their group of symmetries.
