Let  $(M,\omega)$  be a $2n$ dimensional symplectic  manifold and  $X$ is  a smooth vector  field on $M$. Consider the following subvector  space of $\chi^{\infty}(M)$: $$S(X)=\{Y\in \chi^{\infty}(M)\mid  X.\omega(X,Y)=nDiv(X)\omega(X,Y)\}$$ Here $Div$ is the  divergence  correspond to the  volum form $\omega^{n}$


This  vector  space  contains the Lie algebra $C(X)=\{Y\in \chi^{\infty}(M)\mid [X,Y]=0\}$.   It also contains the  Lie  algebra $M(X)=\{fX\mid f\in C^{\infty}(M)\}$

Note that, according to the above definition of $S(X)$,  the inclusion $C(X)\subset S(X)$  sensitivly depends on the scalar $n$.  If we replace $n$ by another scalar, this inclusion is no longer true.( However the inclusion $M(X)\subset S(X)$ is not sensitive to this scalar, it valids  for every other scalar)

**Questions:**
>What other interesting Lie  algebras are  contained in $S(X)$?


>Is $S(X)$ a Li subalgebra of $\chi^{\infty}(M)$?
> If the  answer is no, is the Lie  algebra  generated by $S(X)$ equal to the lie  algebra generated by $C(X)$  and $M(X)$?

>Motivated by the usual dynamical  question "Is the triviality of  centralizer a  generic situation?",  we  ask that: Is it true to say that for  a generic  vector  field $X$  we have $S(X)=M(X)$?


**Note: At the international workshop on dynamical system in ICTP, Italy, 2001, I heared from a specialist of dynamical system that "However the centralizer problem has various aspects both in discret and continuous dynamics, but I think that the symplectic version of this problem is interesting and unknown". So this my post is  a try for a possible symplectization of "centralizer problem"**