\(C^\infty\)-symmetries and non-solvable symmetry algebras (Q2758210)

In the previous article [ibid. 66, No. 2, 111-125 (2001; Zbl 1065.34006)], the authors introduced the notion of \(C^\infty\)-symmetry. This is a generalized form of symmetry depending on a \(C^\infty\)-function \(\lambda\) on jet variables, it is expressed in terms of generators with prolongation formulas, and for \(\lambda\equiv 0\) one gets the usual point symmetries.NEWLINENEWLINENEWLINEHere, an essential use of such symmetries is demonstrated to avoid ``lost symmetries'' in reduction processes in some cases, which is present in the classical way for instance for ODEs with \(\text{SL}(2,\mathbb{R})\) symmetry. Various cases with corresponding strategies are discussed, applications to third-order equations, especially for the Chazy equations, are presented.