Some virtually abelian subgroups of the group of analytic symplectic diffeomorphisms of $S^2$

We show that if $M$ is a compact oriented surface of genus 0 and $G$ is a subgroup of ${\text{Symp}}_{\mu }^{\omega }(M)$ which has an infinite normal solvable subgroup, then $G$ is virtually abelian. In particular the centralizer of an infinite order $f\in {\text{Symp}}_{\mu }^{\omega }(M)$ is virtually abelian. Another immediate corollary is that if $G$ is a solvable subgroup of ${\text{Symp}}_{\mu }^{\omega }(M)$ then $G$ is virtually abelian. We also prove a special case of the Tits Alternative for subgroups of ${\text{Symp}}_{\mu }^{\omega }(S^2)\mathrm{.}$