# An algebraics Hamiltonian vector field with a finite number of periodic orbits(2)

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