We show that any analytically integrable Hamiltonian system near an equilibrium point admits a convergent Birkhoﬀ normalization. The proof is based on a new, geometric approach to the topic. 1. Introduction Among the fundamental problems concerning analytic (real or complex) Hamiltonian systems near an equilibrium point, one may mention the following two: 1) Convergent Birkhoﬀ. In this paper, by “convergent Birkhoﬀ” we mean a normalization, ., a local analytic symplectic system of coordinates in which the Hamiltonian function will Poisson commute with the semisimple part of its quadratic part. .