Let $P:S^{3}\to S^{2}$ be the Hopf fibration. For a vector field $X$ on $S^{2}$ there is a non vanishing vector field $\tilde{X}$ on $S^{3}$ such that $DP(\tilde{X})=X$. It is constructed in two steps:

1)(I thank D. Panazzolo for this first step) We choose a 2 dim subbundle $E$ of $TS^{3}$ which is transverse to one dimensional $\ker DP$. Then $DP:E_{x}\to T_{P(x)}S^{2} $ is a linear isomorphism. Then $DP^{-1}(X)$ is a tangent vector field on $S^{3}$ which vanish on $P^{-1}(\text{singularities of X})$

2)We chose a non vanishing vector field $Z$ on $S^{3}$ tangent to the Hopf fibration. Then $\tilde{X}=DP^{-1}(X)+Z$ is the desired vector field which gives us a one dimensional foliation of $S^{3}$ which leaves are mapped by $P$ to the solutions of $X$ on $S^{2}$.

On the other hand, we know that the singularities are the main obstructions in the study of limit cycles. So the above constructions is a motivation to ask the following questions:

1.Let $Y$ be a non vanishing analytic vec. field on $S^{3}$. Is it true to say that $Y$ has only a finite number of invariant attractor torus?

2.From the view of limit cycle theory, Is it usefull to study the (various) non vanishing lifting of quadratic systems, as non vanishing vec. fields on $S^{3}$? By quadratic system I mean the Poincare compactification of a $2$ degree planar polynomial vector field $$ \begin{cases} \dot x=ax+by+cx^{2}+dxy+ey^{2}\\\dot y=a'x+b'y+c'x^{2}+d'xy+e'y^{2} \end{cases}$$

3.Can we lift an algebraic vector field to a non vanishing algebraic vec. field? An algebraic vec. field on $S^{3}$ is the Poincare compactification of a polynomial vector field on $\mathbb{R}^{3}$.

Finally, it is natural to ask: Is the hopf fibration the only $S^{1}$ fibre bundle from $S^{3}$ to $S^{2}$? (Up to equivalency of fibre bundle)