Lie point symmetries of delay ordinary differential equations

Lie point symmetries of delay ordinary differential equations (DODEs) accompanied by an equation for the delay parameter (delay relation) are considered. A subset of such systems (delay ordinary differential systems or DODSs) which consists of linear DODEs and solution independent delay relations have infinite-dimensional symmetry algebras, as do nonlinear ones that are linearizable by an invertible transformation of variables. Moreover, the symmetry algebras of these linear or linearizable DODSs of order N contain a subalgebra of dimension dim L = 2N realized by linearly connected vector fields. Genuinely nonlinear DODSs of order N have symmetry algebras of dimension n, 0 ≤ n ≤ 2N + 2. It is shown how exact analytical solutions of invariant DODSs can be obtained using symmetry reduction. In particular we present invariant solutions of a DODS originating in a study of traffic flow.