Well, you only need to construct a Hamiltonian torus action with isolated fixed points for that...

@JoseAntonioVallejo Thank you very much for your attention to my question.I appreciate your interesting approach of Hamiltonian action of torus.

By periodic orbit I mean a NON SIngular solution $\gamma(t)$ which satisfies $\gamma(t+T)=\gamma(t)$ for a fixed period T and all t.

Otherwise we could find an algebraic Hamiltonian H:R^4 \to R with a finite number of critical points. These critical points corresponds to singularities of vector field. X_H.

Moreover, according to your interesting approach of "Hamiltonian action", for every invariant vector field on torus we would obtain a Hamiltonian vector field on R^4. But notice that every invariant vector field on torus has either no periodic orbit or infinite number of periodic orbit(rational or irrational slop foliation of torus). This obviously conflict to what we desire.

Am I mistaken?

By the way, Are there some torus action on R^4 with more than "one singularitis" that is an action different from the standard one: (z,w).(x,y)=(z^n.xw^m.y)?

José Antonio Vallejo Just two questions came to my mind: Is every Hamiltonian torus action (which is also analytic and effective) equivariantly equivalents to the standard actions I mentioned in my previous message? As another question: To what extent the moment map of these actions are studied? Are they easy to classify?

The procedure is not really as straightforward as you present it. Given an element of the torus, you generate a (family of) diffeomorphism of the 4-dimensional manifold (this is not restricted to R^4), and then the vector field from that.

Regarding your questions:

1. Not every Hamiltonian torus action is equivalent to the standard one.

2. There is a vast (really vast!) literature on their moment maps. I would suggest to begin with the book "The Topology of Torus Actions on Symplectic Manifolds" by M. Audin (Birkhauser). All the properties I have mentioned can be found there.

Also, a good introduction to Delzant polytopes (which are used in the classification of torus actions) can be seen in the book by Anna Cannas Da Silva, "Lectures on Symplectic Geometry", which is available on-line: https://people.math.ethz.ch/~acannas/Papers/lsg.pdf

Thank you for the link.

BTW did you read my previous comment " By periodic orbit I mean a NON SIngular solution $\gamma(t)$ which satisfies $\gamma(t+T)=\gamma(t)$ for a fixed period T and all"?

Am I mistaken to think you are probably confusing singularities of a vector field with non trivial periodic orbit?

I do not see how does your answers give a complete answer to my question

Is there an algebraic (that is polynomial) H on R^4 such that the corresponding Hamiltonian vector field X_H(which is obviousely a polynomial vector field on R^4) with a finit number of periodic orbits"

I would appreciate if you say me what part of the reference you linked contains an answer to my question.

Let (M,w) be a symplectic manifold and let H be a time dependent Hamiltonian such that it is 1-periodic as a function of t. Consider the Hamiltonian vector field XH determined by H and w, and let f its associated time=1 diffeomorphism on M. Then there is a one-to-one correspondence between k-periodic points of f and k-periodic orbits of XH. My comment simply states that any manifold admitting a Hamiltonian torus action with isolated fixed points, also admits a Hamiltonian diffeomorphism with finitely many periodic points. Thus, in order to get your example you have to construct a Hamiltonian torus action on R^4 with isolated fixed points. The techniques for doing that are explained in the books I mentioned.

José Antonio Vallejo However the subject of time dependent Hamiltonians is an interesting subject, but in reality in my question i am exclusively considering the autonomus case: time independent.(Please see the MO link of my main question).After we remove singularities of X_H then we have a foliation question: No matter of parametrization. we require that the number of closed leaves of the foliation would be finite but non zero.

I realy appreciate your attention and your attempt, but I am not sure your answer can provide a complete answer to my question.

Kindly notice that in the construction described above, you end up by fixing t=1, that is, with a standard time-independent Hamiltonian admitting a finite number of periodic orbits (that is, trajectories c:R -> M satisfying c(t+T)=c(t) for some period T). I think this was your original question.

However, constructing a concrete example may be not straightforward. I learned this technique from some works by V. Ginzburg related to the Conley conjecture, and it is theoretically plausible, but I must admit that I have spent some time trying to figure out a simple example in R^4 with no success. It is just an idea in case you have time and energy to pursue it.