Nonlocal representation of the sl(2,R) algebra for the Chazy equation

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 nd 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 Painleve series. The singularity analysis is applied to the new third-order equation and we nd that it admits two solutions, one given by a left Painleve series and one given by a right Painleve series where the leading-order behaviors and the resonances are explicitly those of the Chazy equation.

Mathematics Subject Classication (2010): 34M25, 34M55, 70G65.

Key words: Chazy equation, Lie symmetries, nonlocal symmetries, singularity analysis.