Suppose $V$ is a no-where zero vector field on $S^n$ ($n$ odd). Let $p \in S^n$. Let $\gamma_p$ be the unique curve on $S^n$ through $p$ and tangential to $V$ everywhere along it. Is it true that $\gamma_p$ is a closed curve $\forall p \in S^n$? If so, is it true that the length of $\gamma_p$ must be finite?

Alternatively, is it possible to find a curve $\gamma$ on $S^n$ of infinate length such that the tangent vectors of $\gamma$ can be extended to form a continuous, no-where zero tangential vector field of $S^n$?