TITLE: Independent vector fields $X,Y$ on $S^3$ with $[X,Y]=Y$
QUESTION [7 upvotes]: Are there two smooth independent vector fields $X,Y$ on $S^3$ with  $[X,Y]=Y$?

REPLY [4 votes]: You can also use this result which appears in the following paper of E. Ghys see corollary I.1.2: Let $G$ be a Lie group homeomorphic to $\mathbb{R}^n$ which acts locally freely on a $n+1$-dimensional manifold $M$. Then the universal cover of $M$ is diffeomorphic to $\mathbb{R}^{n+1}$.
The vector field $X,Y$ define an action of $Aff(\mathbb{R})$ on $S^3$, but the universal cover of $S^3$ is $S^3$ and not $\mathbb{R}^3$.
http://perso.ens-lyon.fr/ghys/articles/actionslocalement.pdf