The Poincare compactification is a method to carry a polynomial vector field on the plane to an analytic vector field on $S^{2}$ via analytic embedding $$(x,y)\to (\frac{x}{\sqrt{1+x^{2}+y^{2}}},\frac{y}{\sqrt{1+x^{2}+y^{2}}},\frac{1}{\sqrt{1+x^{2}+y^{2} }})$$ which maps $\mathbb{R}^{2}$ to the upper hemi sphere. Similarly we map the plane to the lower hemi sphere. We denote this embeding by $\phi$. If $X$ is a polynomial vector field of degree $n$, then $z^{n-1}\phi_{*} X$ is an analytic vector field on $S^{2}$.
Now consider a polynomial vector field $X$ on the plane with a unique singularity at the origin,for example the Lienard equation.
Is there an analytic embedding $\psi$ from $\mathbb{R}^{2}-\{0\}$ onto $\mathbb{T}^{2}-S^{1}$ and an analytic function $g:\mathbb{T}^{2} \to \mathbb{R}$, $g$ vanish at $S^{1}$, such that $g\psi_{*} X$ is an analytic vector field on torus?
What is a possible obstruction for such torus compactification?
