. Proof, From the expression above, we can see? i(2j?1) (z) and? i(2j) (z) have conjugate values. Indeed, according to the properties of ? and? i above, we have 2l, any element ? := (? 1, ? n ) ? R m (cf. (2.1)), we have ? ? := (? ?(1) , . . . , ? ?(n) ) ? R ?(m) . It follows that if? im,? w ? is a resonant term in? im (w), then? im

, z 2l?1 , z 2l =z 2l?1 , z 2l+1 , . . . , z n ) and for m > 2l, the values (not the functions)? im (z 1 ,z 1 , . . . , z 2l?1 ,z 2l?1 , z 2l+1, Hence, for m 2l,? im and? i?(m) are a pair of conjugate functions of variables, vol.2

=. , .. .. , and ,. .. , F q ) be a formal non-degenerate discrete integrable system of type (p, q) on R n at a common fixed point, say the origin 0. We assume that the family of its linear parts {A j x} is either projectively hyperbolic or infinitesimally integrable with a weakly non-resonant family of generators. Assume furthermore that the commuting family of real diffeomorphisms {? i } satisfies A j ? i = ? i A j

P. , .. .. P-?1-?-p, P. , P. , and .. , F q ? P ) satisfies assumption of Theorem 2.6, then P ?1 ? i P is of the form (2.6) with (2.7), for all i. Therefore, Hence, the family {P ?1 ? i P } is in Poincaré-Dulac normal form as it commutes with the family of its linear part {D j }Since the family (P ?1 ? 1

V. I. Arnold, Mathematical methods of classical mechanics, Graduate Texts in Mathematics, vol.60, 1989.

V. I. Arnold, Geometrical methods in the theory of ordinary differential equations, Grundlehren der Mathematischen Wissenschaften, vol.250, 1988.

O. I. Bogoyavlenskij, Extended integrability and bi-Hamiltonian systems, Communications in mathematical physics, vol.196, issue.1, pp.19-51, 1998.

D. Bambusi and L. Stolovitch, Convergence to normal forms of integrable PDEs, p.29, 2020.
URL : https://hal.archives-ouvertes.fr/hal-02094584

M. and C. , Géométrie différentielle et singularités de systèmes dynamiques, vol.444, pp.138-139, 1986.

M. Chaperon, A Forgotten Theorem on Z k × R m -action Germs and Related Questions, Regular and Chaotic Dynamics, vol.18, issue.6, pp.742-773, 2013.

W. Culver, On the existence and uniqueness of the real logarithm of a matrix, Proceedings of the American Mathematical Society, vol.17, issue.5, pp.1146-1151, 1966.

H. Eliasson, Hamiltonian systems with Poisson commuting integrals, 1984.

H. Eliasson, Normal forms for Hamiltonian systems with Poisson commuting integrals-elliptic case, Commentarii Mathematici Helvetici, vol.65, issue.1, pp.4-35, 1990.

F. R. Gantmakher, The theory of matrices, 1959.

X. Gong and L. Stolovitch, Real submanifolds of maximum complex tangent space at a CR singular point, I, Invent. math, vol.206, pp.293-377, 2016.

H. Ito, Convergence of Birkhoff normal forms for integrable systems, vol.64, pp.412-461, 1989.

H. Ito, Integrability of Hamiltonian systems and Birkhoff normal forms in the simple resonance case, Mathematische Annalen, vol.292, issue.1, pp.411-444, 1992.

K. Jiang, Local normal forms of smooth weakly hyperbolic integrable systems, Regular and Chaotic Dynamics, vol.21, issue.1, pp.18-23, 2016.

T. Kappeler, J. Pöschel, &. Kdv, and . Kam, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol.45

. Springer-verlag, , 2003.

S. Kuksin and G. Perelman, Vey theorem in infinite dimensions and its application to KdV, Discrete Contin. Dyn. Syst, vol.27, issue.1, pp.1-24, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00832747

W. Li, J. Llibre, and X. Zhang, Extension of Floquet's Theory to Nonlinear Periodic Differential Systems and Embedding Diffeomorphisms in Differential Flows, American Journal of Mathematics, vol.124, issue.1, pp.107-127, 2002.

J. Liouville, Note sur l'intégration deséquations différentielles de la Dynamique, Journal de Mathématiques Pures et Appliquées, pp.137-138, 1855.

L. Stolovitch, Singular complete integrability, Publications Mathématiques de l'Institut des HautesÉtudes Scientifiques, vol.91, pp.133-210, 2000.

L. Stolovitch, Normalisation holomorphe d'algèbres de type Cartan de champs de vecteurs holomorphes singuliers, Annals of mathematics, pp.589-612, 2005.

L. Stolovitch, Family of intersecting totally real manifolds of (C n , 0) and germs of holomorphic diffeomorphisms, vol.143, pp.247-263, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01285115

A. P. Veselov, Integrable mappings, Uspekhi Mat. Nauk, vol.46, pp.3-45, 1991.

A. P. Veselov, Growth and integrability in the dynamics of mappings, Comm. Math. Phys, vol.145, issue.1, pp.181-193, 1992.

J. Vey, Sur certains systemes dynamiques séparables, American journal of mathematics, vol.100, pp.591-614, 1978.

S. Walcher, On differential equations in normal form, Mathematische Annalen, vol.291, issue.1, pp.293-314, 1991.

X. Zhang, Analytic normalization of analytic integrable systems and the embedding flows, Journal of Differential Equations, vol.244, issue.5, pp.1080-1092, 2008.

X. Zhang, Analytic integrable systems: Analytic normalization and embedding flows, Journal of Differential Equations, vol.254, issue.7, pp.3000-3022, 2013.

S. L. Ziglin, Branching of solutions and nonexistence of first integrals in Hamiltonian mechanics. I, Functional Analysis and Its Applications, vol.13, pp.181-189, 1982.

N. T. Zung, Convergence versus integrability in Birkhoff normal form, pp.141-156, 2005.

N. T. Zung, Non-degenerate singularities of integrable dynamical systems, Ergodic Theory and Dynamical Systems, vol.35, pp.994-1008, 2015.
URL : https://hal.archives-ouvertes.fr/hal-00999733

N. T. Zung, A conceptual approach to the problem of action-angle variables, Archive for Rational Mechanics and Analysis, vol.229, issue.2, pp.789-833, 2018.

, UMR CNRS 7351 Université de Nice Sophia-Antipolis E-mail address: kai.jiang@bicmr.pku.edu.cn CNRS and