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Consider an analytic Hamiltonian system near its analytic invariant torus$\mathcal T_0$ carrying zero frequency. We assume that the Birkhoff normal formof 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 thatin this case there is an analytic canonical transformation -- not just a formalpower series -- bringing the Hamiltonian into its Birkhoff normal form.

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