ODEs with essential contact or point symmetries

(joint work with Eivind Schneider)
We observe that, up to conjugation, a majority of higher order ODEs and ODE systems have only point fiber-preserving symmetries (surprisingly this is also true for "most interesting" ODEs). We describe all the exceptions in the case of scal ar ODEs and systems of pairs of ODEs on a pair of functions. We exploit classifications of Lie algebras of vector fields in 2 and 3 dimensions.
While we can express scalar ODEs with essentially contact or point symmetry algebras via absolute and relative differential invariants, we have to invoke also conditional differential invariants in the case of ODE systems to deal with singu lar orbits of the action. In the scalar case the result is partially due to Lie, but we consider the global classification and discuss the algebra of relative invariants. For systems the result is new.
Investigating prolongations of the actions, we observe some interesting relations between different realizations of Lie algebras. We also note that prolongation of a finite-dimensional Lie algebra acting on a differential equation may not eventually become free. An example of underdetermined ODE with this phenomenon shows limitations of the method of moving frames.