Symmetry and the Chazy Equation

There are three different actions of the unimodular Lie group SL(2, ) on a two-dimensional space. In every case, we show how an ordinary differential equation admitting SL(2) as a symmetry group can be reduced in order by three, and the solution recovered from that of the reduced equation via a pair of quadratures and the solution to a linear second order equation. A particular example is the Chazy equation, whose general solution can be expressed as a ratio of two solutions to a hypergeometric equation. The reduction method leads to an alternative formula in terms of solutions to the Lamé equation, resulting in a surprising transformation between the Lamé and hypergeometric equations. Finally, we discuss the Painlevé analysis of the singularities of solutions to the Chazy equation