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


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