2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/58612We 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.6 pages, minor revisions in second versonDynamical SystemsGeometric Topology58C30; 22E25text