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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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