Let $X$ be a polynomial vector field of degree $2$ on $\mathbb{R}^{2}$. Does there exist a nonvanishing smooth function $g$ such that $Div(gX)$ is a proper map?Or at least the zero locus of $Div(gX)=0$ is a compact set? Can we find this $g$ an algebraic map or at least in the form $e^{P(x,y)}$ where $P$ is a polynomial?

Is there a uniform upper bound $PDH(n)$ for the number of limit cycles of a those polynomial vector field of degree $n$ for which the divergence is proper or at least $Div=0$ is a compact curve. For $n=3$, two what extent thses vector field are classified in term of their coefficients