  We observe that, up to conjugation, a majority of symmetric higher order ODEs
(ordinary differential equations) and ODE systems have only fiber-preserving
point symmetries. By exploiting Lie's classification of Lie algebras of vector
fields, we describe all the exceptions to this in the case of scalar ODEs and
systems of ODEs on a pair of functions.
  The scalar ODEs whose symmetry algebra is not fiber preserving can be
expressed via absolute and relative scalar differential invariants, while a
similar description for ODE systems requires us to also invoke conditional
differential invariants and vector-valued relative invariants to deal with
singular orbits of the action.
  Investigating prolongations of the actions, we observe some interesting
relations between different realizations of Lie algebras. We also note that it
may happen that the prolongation of a finite-dimensional Lie algebra acting on
a differential equation never becomes free. An example of an underdetermined
ODE system for which this phenomenon occurs shows limitations of the method of
moving frames.
