# Nice Form of Vector Field

Let $$G$$ be a reductive algebraic group (maybe reductive is not necessary) over an algebraically closed field $$k$$ of characteristic zero. Let $$X$$ be a homogeneous affine $$G$$-variety, i.e. $$X=G/K$$ for some reductive subgroup $$K$$ of $$G$$.

In this case we obtain an action by vector fields of the Lie algebra $$\mathfrak{g}$$ of $$G$$ on functions on $$X$$, the algebra $$k[X]$$. Choose an element $$u\in\mathfrak{g}$$.

My question is: are there statements about nice local forms of $$u$$? e.g., I would like to know about statements of the form: If $$u$$ vanishes at a point $$a$$ on $$X$$ there exists a neighborhood $$U$$ of $$a$$ with coordinates $$x_1,\dots,x_n$$ (coming from an etale map $$U\to\mathbb{A}^n$$) such that $$u=\sum\limits_i x_{j_i}\partial_{x_{k_i}}$$ Any references or comments are greatly appreciated!

Let's look on the linearization of the action (it exists for a lie group $$G$$ acting on affine variety $$X$$). That way, $$X$$ is embedded into an affine space $$V = \mathbb{A}^N$$ as a closed subset, and $$G$$ acts linearly on $$V$$. Consider now an element $$s \in \mathfrak{g}$$ and the corresponding linear vector field on $$V$$, let $$x \in X$$ be its fixed point. $$T_X(x)$$ is an invariant subspace of $$s$$. Provided it would have an invariant complement, projection along this complement would give us the desired form.
For reductive $$K$$ we can do even better - $$Stab_x \cong K$$, and $$K$$ is reductive, so we can chose the invariant complement with respect to whole $$K$$. Projection along this complement on $$T_x(X)$$ gives the desired etale linearization for the whole stabilizer simultaneously.
EDIT: actually I've realised that you ask about somewhat strange normal form. I read it automatically as linear vector field ($$\Sigma a_{ij}x^i \partial_j$$). The rest is linear algebra anyways (the form you are demanding as it is written I think does not exist even for linear vector fields).