Is  there  a  non vanishing  vector  field on $S^{3}$ with  an infinite  family $T_{\lambda}$ of  invariant  torus such that each $T_{\lambda}$ has  an  structure of  a  Kronecker  foliation but for every two  different $\lambda_{1}$ and $\lambda_{2}$, the  corresponding  Kronecker  foliations are non topological equivalent?


Is  there an  example  of a non vanishing  vector  field $\tilde{X}$on $S^{3}$  with the  above  property with the  additional  condition that $\tilde{X}$ is  a  lifting of  a  vector  field  $X$ on $S^{2}$  with  a  center singularity  via  Hopf  fibration, see [this related question](https://mathoverflow.net/questions/182139/lifting-a-quadratic-system-to-a-non-vanishing-vector-field-on-s3).