Let  $p:S^3 \to S^2$ be the Hopf fibration  which is  a  result of the standard action of $S^1$ on $S^3$.

>Is there a  $2$  dimensional vector  bundle $\tilde{p}:E \to S^2$ such that $S^3\subset E$ and $\tilde{p}_{|S^3}=p$?
>
>Is there a vector bundle  $E$ as above with the following extra  condition:The total space  $E$ can be acted by $S^1$ with linear isomorphism and this action would be the extension of the standard action of $S^1$ on $S^3$?

**Note:** One can ask the same question for  such extension of an  arbitrary principal bundle  to  an equivariant vector bundle of arbitrary dimension.