Stability of equilibrium solutions of Hamiltonian systems under the presence of a single resonance in the non-diagonalizable case

The problem of knowing the stability of one equilibrium solution of an analytic autonomous Hamiltonian system in a neighborhood of the equilibrium point in the case where all eigenvalues are pure imaginary and the matrix of the linearized system is non-diagonalizable is considered. We give information about the stability of the equilibrium solution of Hamiltonian systems with two degrees of freedom in the critical case. We make a partial generalization of the results to Hamiltonian systems with n degrees of freedom, in particular, this generalization includes those in 1].