Automorphisms and transformations of solutions to the generalised Chazy equation for various parameters

We analyse the automorphisms of solutions to Chazy’s equation and the generalised Chazy equation for the parameters $k=2,3,4$ and $9$. These automorphisms are induced by triangular domains with isosceles symmetry. We also prove theorems about the transformations of solutions to the generalised Chazy equation between various different parameters. Using the transformation of solutions between parameters $k=2$ and $k=\frac{2}{3}$, we are able to prove a result about the automorphism of the solutions to the $k=\frac{2}{3}$ generalised Chazy equation.