Convergence of the Birkhoff normal form sometimes implies convergence of a normalizing transformation

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.