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, et al.).
  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](http://www.mathunion.org/ICM/ICM1990.2/Main/icm1990.2.1259.1270.ocr.pdf) 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 1:** It would be interesting to look at the behavior of $Div(\tilde{X})$, since this divergence must vanish somewhere between two invariant torus which  covers two limit cycles $\gamma_{1}$ inside $\gamma_{2}$. This  is  a motivation to ask: Can we obtain a lifting $\tilde{X}$ which divergence is constant on each fibre?Even if the  answer is negative  we still have a vanishing result  for  some  quantity associated to  a vector field on $S^{2}$. The latter statement would be more clear after looking at some thing similar to [Dulac-Bendixon criterion](http://en.wikipedia.org/wiki/Bendixson%E2%80%93Dulac_theorem) 

>**Note2:** One  can repeat the same initial question of nonvanishing  lifting vector field for $\pi:TS^{2}\to S^{2}$ where $\pi$ is the natural projecting map(Or lifting the vector fields on $S^{2}$  via circle bundle $P:T^{1} S^{2}\to S^{2}$, where $T^{1}S^{2}$ is the unit tangent bundle).  For this type of lifting we possibly  need  to the answer of [this question](http://mathoverflow.net/questions/186401/a-question-on-tangent-bundle-and-second-tangent-bundle?lq=1)