  We show the existence of an anti-pluricanonical curve on every smooth
projective rational surface X which has an infinite group G of automorphisms of
either null entropy or of type Z . Z (semi-direct product), provided that the
pair (X, G) is minimal. This was conjectured by Curtis T. McMullen (2005) and
further traced back to Marat Gizatullin and Brian Harbourne (1987). We also
prove (perhaps) the strongest form of the famous Tits alternative theorem.
