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In the geometry of generic 2-plane fields on 5-manifolds, the localequivalence problem was solved by Cartan who also constructed the fundamentalcurvature invariant. For generic 2-plane fields or $(2,3,5)$-distributionsdetermined by a single function of the form $F(q)$, the vanishing condition forthe curvature invariant is given by a 6$^{\rm th}$ order nonlinear ODE.Furthermore, An and Nurowski showed that this ODE is the Legendre transform ofthe 7$^{\rm th}$ order nonlinear ODE described in Dunajski and Sokolov. We showthat the 6$^{\rm th}$ order ODE can be reduced to a 3$^{\rm rd}$ ordernonlinear ODE that is a generalised Chazy equation. The 7$^{\rm th}$ order ODEcan similarly be reduced to another generalised Chazy equation, which has itsChazy parameter given by the reciprocal of the former. As a consequence ofsolving the related generalised Chazy equations, we obtain additional examplesof flat $(2,3,5)$-distributions not of the form $F(q)=q^m$. We also give4-dimensional split signature metrics where their twistor distributions via theAn-Nurowski construction have split $G_2$ as their group of symmetries.

Flat $(2,3,5)$-Distributions and Chazy's Equations