C-infinity-symmetries and non-solvable symmetry algebras

If an ordinary differential equation admits the non-solvable Lie algebra sl (2, R) as symmetry algebra then, when the classical Lie method of reduction is applied, at least one of its generators cannot be used to obtain a seco nd order reduction. In this paper it is proved that these generators can be recovered as C-infinity-symmetries of the reduced equations. These C-infin ity-symmetries can be used to new-order reductions if the order of the last reduced equations is higher than one. As a consequence, a classification o f the third-order equations that admit sl(2, R) as symmetry algebra is give n. This step by step method of reduction is applied to the Chazy equation.