I'm interested in finding sufficient conditions for when a vector field $X$ defined on an open subset of a smooth manifold $M$ can be extended. It is clear that this can't always be done. For example $X(t) = \frac{1}{t}\frac{\partial}{\partial t}$ on $(0,\infty)$ admits no continuous extension on $\mathbb{R}$.
In particular let's consider a manifold $M = (0,\infty) \times \mathbb{R}^2$ with coordinates $(t,r,\phi)$ where $(r, \phi)$ are the usual spherical coordinates for $\mathbb{R}^2\setminus \{0\}$. I want to prove that if $\widetilde{M}$ is an extension of $M$, then $\frac{\partial}{\partial \phi}$ can be extended onto $\widetilde{M}$. An instance where this occurs is in the Milne Universe. Any help is appreciated!