Linearization of vector fields 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.
 A: Since you don't say what your 'some other conditions are', it is hard to know what result you are claiming.  However, it is not difficult to show that, for $\mathbb{R}^4$ endowed with the symplectic form $\omega = dp\wedge dx + dq\wedge dy$, the vector field
$$
Z = x\ \partial_x + (2y{+}x^2)\ \partial_y - (p{+}2xq)\ \partial_p - 2q\ \partial_q
$$
is Hamiltonian and yet it cannot be smoothly (let alone real-analytically) linearized near the origin.  The reason is the presence of $C^1$ integral curves of $Z$ passing through the origin that are not $C^2$, let alone smooth.  For example, the curve $(x,y,p,q)=(x,x^2\ln|x|,0,0)$ is an integral curve of $Z$.  (For the linearized vector field (which is also Hamiltonian)
$$
Z_0 = x\ \partial_x + 2y\ \partial_y - p\ \partial_p - 2q\ \partial_q\ ,
$$
every integral curve passing through the origin is smooth.)
A: The result of Poincaré is in fact not about the real case but about the complex case. Suppose a vector field of the form
$$ X = \sum_i\lambda_i x^i \partial_i + \text{higher order terms}$$
Then there is an analytic linearization diffeomorphism in a neighborhood of the origin, provided
$$ \text{all} ~\lambda_i~ \text{lie in the same half plane about the origin}$$
and
$$ \lambda_i \neq \sum_j m_i  \lambda_j ~ \text{for any non-negativ integral} ~m_i~ \text{such that} \sum_i m_i> 1.$$
This result is cited in "Local Contractions and a Theorem of Poincaré" by Shlomo Sternberg.
Note that the semisimplicity of your matrix is not sufficient; I think you cannot get away without some diophantic conditions on the eigenvalues as stated above. This maybe seems somewhat strange at first, but it is not hard to construct counterexamples in the other case.
A: It might be worth mentionning works on linearization/normal-forms of Poisson structures, in particular around a degenerate singular point.


*

*Formes normales de structures de Poisson ayant un 1-jet nul en un point (in French) for the formal approach, by Dufour and Wade

*Classification analytique de structures de Poisson (in French) for the analytic approach, by Lohrmann
See also other works from these authors.
