### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 029, 28 pages      arXiv:1506.02473      https://doi.org/10.3842/SIGMA.2016.029

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

Matthew Randall
Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlářská 2, 611 37 Brno, Czech Republic

Received September 23, 2015, in final form March 14, 2016; Published online March 18, 2016

Abstract
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.

Key words: generic rank two distribution in dimension five; conformal geometry; Chazy's equations.

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