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.