  Differential invariants of a (pseudo)group action can vary when restricted to
invariant submanifolds (differential equations). The algebra is still governed
by the Lie-Tresse theorem, but may change a lot. We describe in details the
case of the motion group $O(n)\ltimes\R^n$ acting on the full (unconstraint)
jet-space as well as on some invariant equations.

Differential Geometry