  We show that every real analytic action of a connected supersoluble Lie group
on a compact surface with nonzero Euler characteristic has a fixed point. This
implies that E. Lima's fixed point free $C^{\infty}$ action on $S^2$ of the
affine group of the line cannot be approximated by analytic actions. An example
is given of an analytic, fixed point free action on $S^2$ of a solvable group
that is not supersoluble.
