Let $F, G:\mathbb{R}^D \times \mathbb{R} \to \mathbb{R}^D$ be time-dependent vector fields (as smooth as you want).
I stumbled over the following partial differential equation:
$$ \frac{\partial F}{\partial t} = [F,G]$$
where $[F,G]$ is the Lie-bracket between $F$ and $G$, thus
$$[F,G] = \nabla_{x}G \cdot F - \nabla_{x} F \cdot G$$
Question: Given $G$, what can we say about $F$?
In particular: I know that $G$ is a conservative vector field, that is $G(x,t)=\nabla_x \phi(x,t)$ for some scalar function $\phi(x,t)$. Does this constraint $F$ to be conservative as well?
Thanks for your attention and interest!