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!