The standard frame for $S^3$ consists of $X_i,X_j,X_k$ with $X_i(a)=ia, X_j(a)=ja, X_k(a)=ka$ where $i,j,k$ are standard quaternion numbers, $a\in S^3$, and the multiplication is the quaternion multiplication. This global frame on $S^3$ gives us a trivialization of $TS^3\simeq S^3\times \mathbb{R}^3$. So the Hopf map $P:S^3\to S^2\subset \mathbb{R}^3$ is counted as a unit vector field on $S^3$. This vector field is called "Hopf vector field".

Does the Hopf vector field have a periodic orbit? Is this vector field discussed in various attempts to find an analytic counter example to Seifert conjecture?