Suppose you have a Vector field $X$ on a smooth (complete) manifold $M$, whose flow $\phi_X^t$ is, for each time $t\in (-\varepsilon,\varepsilon)$, smooth (of class $C^k$). Questions: Is $X$ smooth (of class $C^k$)? What about the case $M=\mathbb{R}^n$? Thank you in advance!