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)