  The framework of Baikov-Gazizov-Ibragimov approximate symmetries has proven
useful for many examples where a small perturbation of an ordinary differential
equation (ODE) destroys its local symmetry group. For the perturbed model, some
of the local symmetries of the unperturbed equation may (or may not) re-appear
as approximate symmetries, and new approximate symmetries can appear.
Approximate symmetries are useful as a tool for the construction of approximate
solutions. We show that for algebraic and first-order differential equations,
to every point symmetry of the unperturbed equation, there corresponds an
approximate point symmetry of the perturbed equation. For second and
higher-order ODEs, this is not the case: some point symmetries of the original
ODE may be unstable, that is, they do not arise in the approximate point
symmetry classification of the perturbed ODE. We show that such unstable point
symmetries correspond to higher-order approximate symmetries of the perturbed
ODE, and can be systematically computed. Two detailed examples, including a
fourth-order nonlinear Boussinesq equation, are presented. Examples of the use
of higher-order approximate symmetries and approximate integrating factors to
obtain approximate solutions of higher-order ODEs are provided.
