Answer
Does not exist.
Work Step by Step
Suppose, we have a vector field $G$ such that $div [curl (G)]=0$
Let us consider $F=ai+b j+c k$
$div F=\dfrac{\partial a}{\partial x}+\dfrac{\partial b}{\partial y}+\dfrac{\partial c}{\partial z}$
Given: $curl G=\lt x \sin y, \cos y, z-xy \gt$
This implies that $div[curl(G)]=\sin y-\sin y+1 $
we can see that $div [curl (G)]=1 \ne 0$
Hence, we can conclude that there does not exist such a vector field $G$.
