Typically a symmetry is taken to be something that leaves the action invariant. However, on a classical level, isn't that asking way too much? To match what we conceptually mean by symmetry, we only need something that maps solutions to solutions, so something which leaves the action invariant in the neighbourhood of extremal solutions(?) (After all who cares what a symmetry operation does to unphysical solutions?). Is it because of Noether's theorem?