Let $M$ be a connected closed smooth manifold of dimension $n$, and let $X$ be a smooth vector field on $M$. Since $M$ is compact, $X$ is complete (i.e., its maximal integral curves are defined on all of $\mathbb{R}$).
Given a smooth map $F:N\to M$, a vector field $\tilde{X}$ on $N$ is said to be $F$-related to $X$ if $TF \circ \tilde{X}= X\circ F$, where $TF:TN\to TM$ is the tangent map of $F$. Let us refer to any such $F$-related vector field $\tilde{X}$ as a lift of $X$ via $F$. If $F$ is a local diffeomorphism, then there is a unique lift $\tilde{X}$ of $X$ via $F$, and it is given by the formula $\tilde{X}(x)=(T_x F)^{-1}X(F(x))$.
Question: does there always exist a surjective local difeomorphism $F:\mathbb{R}^n\to M$ such that the unique lift $\tilde{X}$ of $X$ to $\mathbb{R}^n$ via $F$ is complete? If not, does there always at least exist a surjective smooth map $F:\mathbb{R}^n \to M$ and a complete lift $\tilde{X}$ to $\mathbb{R}^n$ of $X$ via $F$?
Motivation: there always exists a surjective local diffeomorphism $F:\mathbb{R}^n\to M$. A construction of this can be given by slightly generalizing one explained by Ryan Budney (at Local diffeomorphism from $\mathbb R^2$ onto $S^2$): construct an injective immersion $\mathbb{R}\to M$ with dense image, then extend this to a local diffeomorphism $$F: \mathbb{R}^n \approx (-\varepsilon, \varepsilon)^{n-1}\times \mathbb{R}\to M$$ by slightly "fattening" the immersion $\mathbb{R}\to M$ in transverse directions. However, this $F$ will not be a covering map in general (and cannot be, if the universal cover of $M$ is not $\mathbb{R}^n$), and in particular the unique lift $\tilde{X}$ to $\mathbb{R}^n$ of $X$ via $F$ need not be complete. But that raises the question of whether it is possible to construct $F$ in some "$X$-adapted" way as to make $\tilde{X}$ complete.