Stability of Equilibrium Points for a Hamiltonian Systems with One Degree of Freedom in One Degenerate Case

This paper concerns with the study of the stability of one equilibrium solution of an autonomous analytic Hamiltonian system in a neighborhood of the equilibrium point with $1$-degree of freedom in the degenerate case $H= q^4+ H_5+ H_6+\cdots$. Our main results complete the study initiated by Markeev in [9].