We say a holomorphic vector field $X$ is tangent to an effective divisor $D$, if $D_Xs=\lambda s$, where $s$ is the determining holomorphic section of the line bundle $L_D$ corresponding to $D$. If $M$ is a Fano manifold and $L_D=-mK_M$ for $m\in \mathbb{Z}^+$, Jian Song and Xiaowei Wang shows that there is no such holomorphic vector field. What if $m\in \mathbb{Q}$?
My question is that for a given holomorphic vector field $X$ whether we can always find an ample line bundle $L$ and an effective divisor $D\in|L|$, such that $X$ is tangent to $D$. Is it always ture that we can find a divisor $D$ and a holomorphic vector $X_D$ such that $X_D$ is tangent to $D$(here $X_D$ is not given)?