Some virtually abelian subgroups of the group of analytic symplectic diffeomorphisms of S²

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We show that if M is a compact oriented surface of genus 0 and G is a subgroup of ωμ(M) which has an infinite normal solvable subgroup, then G is virtually abelian. In particular the centralizer of an infinite order f ∈ ωμ(M) is virtually abelian. Another immediate corollary is that if G is a solvable subgroup of ωμ(M) then G is virtually abelian. We also prove a special case of the Tits Alternative for subgroups of ωμ(S²).