Suppose $M$ is manifold with a global non-vanishing vector field $X$.
Let $\{U\}_{\alpha\in I}$ be a locally finite covering of $M$ such that for each $\alpha$ and $p\in U_{\alpha}$ the maximal integral curve of $X$ containing $p$ is contained in $U_{\alpha.}$ Is it possible to find a partition of unity $\{\phi_{\alpha}\}_{\alpha\in I}$ subordinate to the covering such that $X(\phi_{\alpha})=0,$ for all $\alpha\in I$?