I have a smooth manifold $M$ with $G$-action and complete intersection of codimension $n$ given by ideal $I$ such that $gI=I$. I'm interested when I can choose a $n$-dimensional vector space of generators of $I$ to be a $G$-representation. Of course, in the case of Lie-algebra action (or non-compact group) I can ask it only in neighborhood of $I$. Is it possible to find example with two distinct subspaces of generators? In analytic case, is it posible that everything is algebraic but such generators transcedental like exponents? There are simple counter-examples for 1-dimensional Lie-algebra action?
Update: it's true for reductive action of $\mathbb{C}^*$ and hypersurface. We can decompose ideal by weights and take component minimal (resp. to corresponding $\mathbb{Z}$-grading) component of hypersurface equation, then it is easy to see that this component generate ideal.
There are counter-examples for codim 2 case? If so, which natural intersections coming with G-representation on generators? Basic example is moment map, what else?