  Quasi-homogeneous surfaces, or Gizatullin surfaces, are normal affine
surfaces such that there exists an open orbit of the automorphism group with a
finite complement. If the action of the automorphism group is transitive, the
surface is called homogeneous. Examples of non-homogeneous Gizatullin surfaces
were constructed in [Ko], but on more restricted conditions. We show that a
similar result holds under less constrained assumptions. Moreover, we exhibit
examples of smooth affine surfaces with a non-transitive action of the
automorphism group whereas the automorphism group is huge. This means that it
is not generated by a countable set of algebraic subgroups and that its
quotient by the (normal) subgroup generated by all algebraic subgroups contains
a free group over an uncountable set of generators.
