In this survey we first summarize the results on the existence of analytic normalization for general analytically integrable systems to their normal forms and for analytically integrable symplectic Hamiltonian systems to their Birkhoff normal forms. Next we summarize the results on the existence of first integrals for analytic differential systems in a neighborhood of a singularity. Also part of them is related to the existence of embedding flows of analytically integrable diffeomorphisms.

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