Nonlocal Representation of the $sl(2,R)$ Algebra for the Chazy equation

TL;DR

This paper explores the nonlocal symmetries of the Chazy Equation, constructs an equivalent third-order differential equation, and finds solutions using singularity analysis and Painlevé series.

Contribution

It introduces a generalized transformation linking the Chazy Equation to a new third-order equation and analyzes its symmetries and solutions.

Findings

Point symmetries become nonlocal for the reduced equation

Constructed an equivalent third-order differential equation

Found solutions with Painlevé Series behaviors

Abstract

A demonstration of how the point symmetries of the Chazy Equation become nonlocal symmetries for the reduced equation is discussed. Moreover we construct an equivalent third-order differential equation which is related to the Chazy Equation under a generalized transformation, and find the point symmetries of the Chazy Equation are generalized symmetries for the new equation. With the use of singularity analysis and a simple coordinate transformation we construct a solution for the Chazy Equation which is given by a Right Painlev\'e Series. The singularity analysis is applied to the new third-order equation and we find that it admits two solutions, one given by a Left Painlev\'e Series and one given by a Right Painlev\'e Series where the leading-order behaviors and the resonances are explicitly those of the Chazy Equation.