KAM-tori Near an Analytic Elliptic Fixed Point

We study the accumulation of an elliptic fixed point of a real analytic Hamiltonian by quasi-periodic invariant tori.
We show that a fixed point with Diophantine frequency vector $\omega_0$ is always accumulated by invariant complex analytic KAM-tori. Indeed, the following alternative holds: If the Birkhoff normal form of the Hamiltonian at the invariant point satisfies a Rüssmann transversality condition, the fixed point is accumulated by real analytic KAM-tori which cover positive Lebesgue measure in the phase space (in this part it suffices to assume that $\omega_0$ has rationally independent coordinates). If the Birkhoff normal form is degenerate, there exists an analytic subvariety of complex dimension at least $d+1$ passing through $0$ that is foliated by complex analytic KAM-tori with frequency $\omega_0$.
This is an extension of previous results obtained in [1] to the case of an elliptic fixed point.