SYMMETRY AND THE CHAZY EQUATION

There are three different actions of the unimodular Lie group SL(2, C) on a two-dimensional space. In every case, we show how an ordinary di fferential equation admitting SL(2) as a symmetry group can be reduced in order by three, and the solution recovered from that of the reduce d equation via a pair of quadratures and the solution to a linear seco nd order equation. A particular example is the Chazy equation, whose g eneral solution can be expressed as a ratio of two solutions to a hype rgeometric equation. The reduction method leads to an alternative form ula in terms of solutions to the Lame equation, resulting in a surpris ing transformation between the Lame and hypergeometric equations. Fina lly, we discuss the Painleve analysis of the singularities of solution s to the Chazy equation. (C) 1996 Academic Press, Inc.