# An algebraic 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?

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