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