HIDDEN AND CONTACT SYMMETRIES OF ORDINARY DIFFERENTIAL-EQUATIONS

The Lie algebra for the maximal contact symmetries of third-order ordi nary differential equations (ODES) is examined for type I and II hidde n symmetries where the analysis of hidden symmetries for point symmetr ies is extended to contact symmetries. ODES invariant under the group associated with the ten-dimensional (maximal) Lie algebra may produce type I hidden symmetries for two-parameter subgroups and type II hidde n symmetries for certain solvable non-Abelian three-parameter subgroup s in the third-order ODES when they are reduced in order. A new class of type II hidden symmetries is recognized in which contact symmetries transform to point symmetries in some reduction paths. Two examples o f ODES invariant under subgroups of the ten-parameter group under whic h y,,, = 0 is invariant demonstrate the new class of type II hidden sy mmetries.