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](https://en.m.wikipedia.org/wiki/Seifert_conjecture)?



Periodic orbit for certain vector field on $S^3$ (à la Seifert conjecture)
I think the question is perhaps confusing, since the term Hopf vector field is usually reserved for your $X\_i$ vector field (which is tangent to the fibers of the standard Hopf fibration). As I understand, you are instead referring to the vector field which, at a point $p$ with $P(p) = (a,b,c) \in S^2 \subset \mathbb{R}^3$, is given by $Y := aX\_i + bX\_j + cX\_k$. Assuming this is the correct interpretation, then the fiber over $(1,0,0)$ has $Y = X\_i$ and so the fiber of $P$ over $(1,0,0)$ is a periodic orbit of $Y$.


