Symmetry and the Chazy Equation
نویسندگان
چکیده
There are three di erent actions of the unimodular Lie group SL on a two dimensional space In every case we show how an ordinary di erential equation admitting SL 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 Lam e equation resulting in a surprising transformation between the Lam e and hypergeometric equations Finally we discuss the Painlev e analysis of the singularities of solutions to the Chazy equation
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