On Stability in Hamiltonian Systems with Two Degrees of Freedom

We consider the stability of the equilibrium position at the origin of coordinates of a Hamiltonian system with two degrees of freedom whose unperturbed part describes oscillators with restoring force of odd order greater than $1$.
It is proved that if the exponents of the restoring force of the oscillators are not equal, then the equilibrium position is Lyapunov stable. If the exponents are equal, then the equilibrium position is conditionally stable for trajectories not belonging to some level surface of the Hamiltonian. The reduction of the system to this surface shows that the equilibrium position is stable in the case of general position.