Simply, we discuss things on $C^{2n}$ ($or R^{2n}$) but not on manifolds. Given an anayltic (or real analytic) vector field $V$ on $C^{2n}$ ($or R^{2n}$), with a zero at the origin . And $V_{0}$ is the linear part of $V$. Suppose one has some other conditions, i.e., usually one of the conditions is that $V_0$ associate with a matrix that is semi-simple. Is there any analytic (real analytic) symplectomorphism $f:(C^{2n},0)\rightarrow (C^{2n},0)$ (or $f:(R^{2n},0)\rightarrow (R^{2n},0)$), such that $f_{*}V=V_0$?

I know that Henri Poincare had the result of linearization of a real analytic vector field on $R^n$, and Sternberg has some theory for linearizing smooth vector fields.

BTW, naturally one can ask the similar question for odd dimensional spaces.

For smooth case of our problem, the answer is yes and I have found some literature. And for (real) analytic case, I guess it also has a well-known result, am I right? Please help me to make sure of it.

Latest update. 13. May. 2013

I found a monograph, "Lectures on Analytic Differential Equations" wrote by Yulij IIyashenko and Sergei Yakowenko, I think it helps me to understand the linearization theory. However, thank you for all your kind helps.

Mathematical Methods of Classical Mechanics. In Appendix 7 he discusses normal forms of Hamiltonian systems in a neighborhood of a stationary point. $\endgroup$ – Liviu Nicolaescu Apr 28 '13 at 13:41