If you are given a symmetry, is there a way to find a non-trivial invariant of that symmetry? e.g. rotational invariance, $x\rightarrow Mx$ you would find the invariant $x^Tx$.
Or if you were given the symmetry of gauge invariance you might find the Maxwell Field invariant. Of coordinate invariance you might find one of the GR invariants.
Is there a general way to find at least one non-trivial invariant, given a symmetry?
And if not, is it provable that there is no general way?