I will discuss two approaches to the equivalence problem:
Lie-Tresse differential invariants and Cartan moving frames. Relation between the two will be clarified.
Invariants derived via Lie equations allow to restrict the symmetry dimensions. There are many examples, including parabolic geometries and Killing tensors (an instance of higher spin fields).
In the non-holonomic situation Lie equations are not in involution, and weighted jets provide a more convenient setup, leading to the Tanaka theory.