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. In fact $\tilde{X}$ 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. (For instance, see the "Non accumulation theorem" by Ilyashenko or the subject of "finite cyclicity of polycycles" by Roussarie Dumortier). 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?(Motivated by "Finiteness theorem for limit cycles" By Ilyashenko)

2.From the view of limit cycle theory, is it usefull to study the (various) lifting of quadratic systems on $s^{2}$ to 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)

Note: One can repeat the same initial question of nonvanishing lifting vector field for $P:TS^{2}\to S^{2}$ where $P$ is the projection map. For this type of lifting we possibly need to the answer of this question