ODE of second order admitting two-dimensional algebras of dynamic symmetries.

A generalization of S.Lie's classification of second order ODEs on two-dimensional algebras of point symmetries is constructed. First integrals for found types second order ODEs are reduced. The possibility of the determination of two-dimensional algebras of dynamic symmetries over number field is considered. Interconnection of dynamic and contact symmetries is demonstrated. On a concrete example it is shown the procedure of the decomposition of a contact transformation into superposition of point transformation and Legendre transformation.