Unstable equilibria of Hamiltonian systems

It is shown that a Hamiltonian system in the neighbourhood of an equilibrium may be given a special normal form in case four of the eigenvalues of the linearized system are of the form $\lambda_1, -\lambda_1, \lambda_2, -\lambda_2,$ with $\lambda_1$ and $\lambda_2$ independent over the reals, i.e., $\lambda_1/\lambda_2 \notin \mathbf R$. That is, for a real Hamiltonian system and concerning the variables $x_1, y_1, x_2, y_2$ the equilibrium is of either type center–saddle or complex–saddle. The normal form exhibits the existence of a four–parameter family of solutions which has been previously investigated by Moser. This paper completes Moser's result in that the convergence of the transformation of the Hamiltonian to a normal form is proven.