C$^{\infty}-$Symmetries and Reduction of Equations Without Lie Point Symmetries

C. Muriel and J. L. Romero

Abstract: It is proved that several usual methods of reduction for ordinary di\-ffe\-ren\-tial equations, that do not come from the Lie theory, are derived from the exis\-ten\-ce of $C^\infty$-sy\-mme\-tries. This kind of sy\-mme\-tries is also applied to obtain two successive reductions of an equation that lacks Lie point sy\-mme\-tries but is a reduced equation of another one with a three dimensional Lie algebra of point symmetries. Some relations between $C^\infty$-sy\-mme\-tries and potential sy\-mme\-tries are also studied.

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