# Vector fields near non defined points

As far as my short knowledge on the analysis of vector fields goes, one is often interested on studying such objects near equilibrium points. So the theory I've read concerns v.f.'s $X$ in a neighbourhood of a point $p$ such that $X(p)=0$. Then one tries to find invariant manifolds which cross at $p$ and which arrange the integral curves of $X$ near $p$.

But what happens if we want to study the v.f. $X(x)$ near a point $p$ such that $\lim_{x\to p}X(x)=\infty$?. Does this even make sense?

For example, I imagine: $X(x)=\frac{f(x)}{|x|^\alpha}\frac{\partial}{\partial x}$, with, say, $\alpha>0$, $|f(0)|=M<\infty$ near $x=0$?

I guess that the theory that states existence and uniqueness of solutions fails. I also know that in the neighbourhood of regular points $X(p)\neq 0$ one may apply the flow-box theorem. But is that possible at infinity?

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