Suppose I am given a holomorphic vector field $X$ over a complex manifold $M$. To simplify this we can suppose that $X$ is a holomorphic vector field in $\mathbb{C}^n$ for $n=2$ or $n=3$. How can I determine another vector field (non-colinear with $X$) such that their Lie Bracker $[X,Y]$=0? I am trying to do this without applying the vector field $[X,Y]$ to an arbitrary $f$ and working with coordinates and a lot of derivatives.
Should this problem be too complicated (and I think it is) we should probably stick with polynomial vector fields (or even with the homogeneous polynomials ones, although these have already been classificated here: link in dimension two).
I am hopeful that there is something done in this sense in some Lie Group theory literature before. Any help is appreciated.
Gustavo, this probably doesn't make you happy, but you need a vector field that's invariant under the flow of $X$. So pick an arbitrary hyperplane $H$ transverse to $X_0=X(0)\in\mathbb C^n$, and let $Y_0\in \mathbb C^n$ be arbitrary. Set $Y(p)=Y_0$ at points $p\in H$. Let $\phi_t$ be the $X$-flow, and take $Y(\phi_t(p)) = (\phi_t)_{*p}Y_0$. Does this work?