As anotnter her possible counter example I would like to pose the following question:

@DanielAsimov On the other hand I am thinking to some other possible counter example: First We know that there are examples which show that Hartman Grobman is valid topologicaly but not in C^1 class(Around singularity). So I wonder can we introduce a vector field around singularity at $0\in \mathbb{R}^4$ which is topologically linearaizable but not smoothly. Then after a Blow up of singularity can we obtain two 0n1 dimensional foliation of $S^3$ which are topological equivalent but not smooth equivalent?

@DanielAsimov Dear Prof. Asimov, Thank you for your attention to my question and your edit. I think a smooth invariant of a Reeb foliation of $S^3$ is the derivative of Holonomy map defined on a 1 dimensional transversal. So If we introduce two different Reeb foliations $F_1, F_2 $ with different $p_1'(0), p_2'(0)$, the derivative of Poincare or Holonomy map so they are not smooth equivalent.

@WillieWong The only thing I had already heard was "Prelman solved Poincare conjecture via Ricci flow" Then I serach for definition of Ricci flow I realized it is a 1 parameter family of metric with certain differential equation. So I posed the question when is this 1 parameter periodic. I was some how inspired by obstructions for manifold to be an Einstein manifold. I thank you again for informing me of the Entropy formula.

@WillieWong Thank you for your comment and Prelman work. I was not aware of Perelman work.

I wonder what is the dynamics and topological behavior of codimension 1 distribution orthogonal to this beautiful example? I would appreciate if you send me an email "[email protected]" for more discussion.

Now that is very great thank you

Thank you and my +1 for your answer. Does not it work for every arbitrary $x_3=c$?So we have infinite number of closed orbits. Am I mistaken?

My +1 and thanks for your attention to my question and your answer. I think some thing is missing. It seems that you actually use the following statement but I think it is not true: "If $N$ is codimension one submanifold and $H_1$ and $H_2$ are constant on $N$ then the two Hamiltonian flow are the same on $N$" . This is not true for example put $N=$ unit circle in the plane, $H_1\equiv 0$ and $H_2=x^2+y^2-1$ The first Hamiltonian consist of singularities and the second one consist of a periodic orbit, two different dynamic. Am I missing some thing?

@PietroMajer Yes I was mistaken since the energy level must have codimension 1. So 3 dimension. Could you please write the Hamiltonian you are talking about

@ThomasRot I think you mean ellipsoid not ellipse. Moreover I think on sphere you have two singularities yes?(not possible qll solutions are closed orbit). I would appreciate if you wrote such a Hamiltonian

@PietroMajer yes that is very good. Thank you

By counter example I mean a connected manifold M with an interval of regular values [a,b] (no level set is empty) but $H^{-1}(a)$ is not diffeomorphic to $H^{-1} (b)$. I mean we emphasis on connected ness of M. (We forget compactness condition). A modified question: a connected manifold and a regular map without critical point which has two non diffeomorphic non empty level set. Thanks again for your attention to my question.