  Consider an analytic Hamiltonian system near its analytic invariant torus
$\mathcal T_0$ carrying zero frequency. We assume that the Birkhoff normal form
of the Hamiltonian at $\mathcal T_0$ is convergent and has a particular form:
it is an analytic function of its non-degenerate quadratic part. We prove that
in this case there is an analytic canonical transformation -- not just a formal
power series -- bringing the Hamiltonian into its Birkhoff normal form.
