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?

**Added September 7, 2020:** What about if we reduce the **polynomial** condition to **Real analytic** condition?

This question is related to my previous quesion