Transcendental Case in Stability Problem of Hamiltonian System with Two Degrees of Freedom in Presence of First Order Resonance

We study Lyapunov stability of an equilibrium for Hamiltonian system with two degrees of freedom. We suppose that the matrix of the linearised system has one pair of zero eigenvalues and its Jordan normal form is not diagonal. We consider the so-called transcendental case when usual technique, based on normalization of the Hamiltonian up to a finite order terms does not give an answer to the stability question. We obtain conditions under which the transcendental case takes place and show that the equilibrium is unstable in the transcendental case. We apply the above results to investigate the stability of conic precession of a symmetric satellite in a circular orbit.