Thank you very much Vaughn for your very interesting answer

Yes I see thanks

Any way I asked the same question from the other answerer and commenter in this post and I did apologized (you and them) already for thiis extra question.

Yes You are right. You did not consider torus knot and I did not say that "The concept of torus knot in your answer'. But the nearby leaves are not embedded plane they are somewhat spiraling around the torus

Thank you very much for your answer. the concept of torus knot remind me of the following questionwhich arose me one week ago: Can we perturbe a torus knot such that the perturbed knot would be included in a (nearby) leaf of the Reeb foliation? My apology for asking this extra question here

Since you talked about the p-q ( torus) knot this remind me of the following questionwhich arose me one week ago: Can we perturbe a torus knot such that the perturbed knot would be included in a (nearby) leaf of the Reeb foliation? My apology for asking this extra question here

@HenrikRüping Since you talked about the torus knot this remind me of the following questionwhich arose me one week ago: Can we perturbe a torus knot such that the perturbed knot would be included in a (nearby) leaf of the Reeb foliation? My apology for asking this extra question here

The 4 dim. vector field is just the vector field we discussed with Henrik in the comments. I did not pay attention that not only torus but also 3 sphere is invariant under the linear vector field.

@HenrikRüping Your idea was perfect please see the answer by David Speyer

Thank you very much and +1 for your perfect answer which perhaps? (I think) is a negative answer to the linked question. So the Torus knot idea of @HenrikRuping was a very good idea

A motivation for this question is the following mathoverflow.net/questions/477495/…

@HenrikRüping Thank you for this comment. But I think that a homemorphic copy of a torus knot has a simple parametrization whose tangent field is a linear vector field on $\mathbb{R}^4$. But my question is about algebraic vector field in $\mathbb{R}^3$.

Do you mean the corresponding concept in sphere

But the relation to horizontal curves in 3 sphere?

Thank you very much for this answer.

@RyanBudney I do not see what is unclear according to you