  It is shown that a Hamiltonian system in the neighbourhood of an equilibrium
may be given a special normal form in case the eigenvalues of the linearized
system satisfy non--resonance conditions of Melnikov's type. The normal form
possesses a two dimensional (local) invariant manifold on which the solutions
are known. If the eigenvalue is pure imaginary then these solutions are the
natural continuation of a normal mode of the linear system. The latter result
was first proved by Lyapounov. The present paper completes Lyapounov's result
in that the convergence of the transformation of the Hamiltonian to a normal
form is proven and the condition that the eigenvalues be pure imaginary is
removed.
