We show that every real analytic action of a connected supersoluble Lie group on a compact surface with non-zero Euler characteristic has a fixed point. This implies that 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.