1) Assume that $E\to  M$ is  a  smooth real vector  bundle and  $\nabla$ is  a   connection. (We  do  not  assume any  metric  compatibility since  we  do not  fix  a  metric  on $E$).  Assume  that  for  every vector  field  $X$ on $M$ with  a  periodic  orbit $\gamma$ and  for every  $s\in \Gamma^{\infty} (E)$, the  section $\nabla_X s$ vanishes on at least one  point of  $\gamma$. Does this  implies  that $E$ is a (trivial) line  bundle?

**The  motivation:** The  Rolle theorem is  not valid in dimension greater than one.

2) Let $\ell$ be  the  canonical  line  bundle on $\mathbb{C}P^{1}\simeq S^2$, thought of as a smooth  complex  line  bundle. Is there  a  connection $\nabla$  such that for  every  section $s$  and  every  periodic  orbit $\gamma$  of  $X$, the  section $\nabla_{X}^{s}$ vanishes  on at least one  point of $\gamma$?

I ask the  above  question because  I searched  for  some  unusual  differential operator  associated with a  vector  field such that  these operators can count the  number  of  attractors of  a  vector  field  $X$. As  a  related post, please see:
https://mathoverflow.net/questions/182415/elliptic-operators-corresponds-to-non-vanishing-vector-fields