Symmetries and conditional symmetries of differential-difference equations

Two different methods of finding Lie point symmetries of differential-difference equations are presented and applied to the two-dimensional Toda lattice. Continuous symmetries are combined with discrete ones to obtain various reductions to lower dimensional equations, in particular, to differential equations of the delay type. The concept of conditional symmetries is extended from purely differential to differential-difference equations and shown to incorporate Bäcklund transformations.