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

TL;DR

This paper studies the symmetries and transformations of solutions to the generalized Chazy equation for specific parameters, revealing automorphisms induced by geometric domains and transformations linking different parameter cases.

Contribution

It introduces new automorphisms and transformations for solutions of the generalized Chazy equation across various parameters, expanding understanding of its solution structure.

Findings

Automorphisms induced by triangular domains with isosceles symmetry.

Transformations between solutions for different parameter values.

Automorphism results for the case k=2/3.

Abstract

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.