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"