C∞−Symmetries and Reduction of Equations Without Lie Point Symmetries

It is proved that several usual methods of reduction for ordinary differential equations, that do not come from the Lie theory, are derived from the existence of C∞ -symmetries. This kind of symmetries is also applied to obtain two successive reductions of an equation that lacks Lie point symmetries but is a reduced equation of another one with a three dimensional Lie algebra of point symmetries. Some relations between C∞ -symmetries and potential symmetries are also studied.