Is there a polynomial Hamiltonian $H:\mathbb{R}^{4}\to \mathbb{R}$ such that the number of nontrivial periodic orbits of the corresponding Hamiltonian vector field $X_{H}$ is finite but different from zero?
Motivation: The motivation comes from the fact that this can not be happen in Hamiltonians with $1$ degree of Freedom, i.e for Hamiltonians $H:\mathbb{R}^2\to \mathbb{R}$.
Added September 7, 2020: What about if we reduce the polynomial condition to Real analytic condition?
This question is related to my previous question
