Given $F = (xe^y, -x \cos z, -ze^y)$, I find that $\operatorname{div}(F) = 0$ thus there should exist a vector field $G$ such that $\operatorname{curl}(G) = F$. I run into issues finding the solution.
To start out I write out $G = (G_1, G_2, G_3)$,
$$\implies xe^y = \frac{\partial G_3}{ \partial y} - \frac{\partial G_2 }{\partial z},$$
$$-x \cos z = -\left( \frac{\partial G_3}{\partial x} - \frac{\partial G_1}{\partial z} \right),$$
and
$$-z e^y = \frac{\partial G_2}{ \partial x} - \frac{\partial G_1}{\partial y}$$
I have a hard time making assumptions at this point since I know that there may be multiple solutions for the vector field $G$. I was hoping I could be pointed in the right direction on what step I should take next.