Hidden and contact symmetries of ordinary differential equations

The Lie algebra for the maximal contact symmetries of third-order ordinary differential equations (ODES) is examined for type I and II hidden symmetries where the analysis of hidden symmetries for point symmetries 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 hidden symmetries for certain solvable non-Abelian three-parameter subgroups 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 of ODES invariant under subgroups of the ten-parameter group under which y,,, = 0 is invariant demonstrate the new class of type II hidden symmetries.