Analytic integrable systems: Analytic normalization and embedding flows

In this paper we mainly study the existence of analytic normalization and the normal form of finite dimensional complete analytic integrable dynamical systems. More details, we will prove that any complete analytic integrable diffeomorphism $F(x)=Bx+f(x)$ in $({\mathbb{C}}^n,0)$ with $B$ having eigenvalues not modulus $1$ and $f(x)=O(\mathrm{\mid }x{\mathrm{\mid }}^2)$ is locally analytically conjugate to its normal form. Meanwhile, we also prove that any complete analytic integrable differential system $\dot{x}=Ax+f(x)$ in $({\mathbb{C}}^n,0)$ with $A$ having nonzero eigenvalues and $f(x)=O(\mathrm{\mid }x{\mathrm{\mid }}^2)$ is locally analytically conjugate to its normal form. Furthermore we will prove that any complete analytic integrable diffeomorphism defined on an analytic manifold can be embedded in a complete analytic integrable flow. We note that parts of our results are the improvement of Moser’s one in {\it Comm. Pure Appl. Math.} 9$($1956$)$, 673–692 and of Poincar'e’s one in {\it Rendiconti del circolo matematico di Palermo} 5$($1897$)$, 193–239. These results also improve the ones in {\it J. Diff. Eqns.} 244$($2008$)$, 1080–1092 in the sense that the linear part of the systems can be nonhyperbolic, and the one in {\it Math. Res. Lett.} 9$($2002$)$, 217–228 in the way that our paper presents the concrete expression of the normal form in a restricted case.