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Schwarz triangle functions play a fundamental role in the solutions of thegeneralised Chazy equation. Chazy has shown that for the parameters $k=2$ and$3$, the equations can be linearised. We determine the Schwarz trianglefunctions that represent the solutions in these cases. Some of these Schwarztriangle functions also show up in the dual cases where $k=\frac{2}{3}$ and$k=\frac{3}{2}$, which suggests intriguing connections between the solutionsfor $k=2$ and $k=3$ with dihedral and tetrahedral symmetry respectively, and$k=\frac{2}{3}, \frac{3}{2}$ with $G_2$ symmetry. Integrating the solutionswhen $k=\frac{2}{3}$, we obtain flat $(2,3,5)$-distributions parametrisedalgebraically by the corresponding Schwarz triangle functions. The Legendredual of these curves are also algebraic, can be computed quite easily and arerelated to the integral curves of the dual generalised Chazy equation withparameter $\frac{3}{2}$.

Schwarz triangle functions and duality for certain parameters of the generalised Chazy equation