  We show that the automorphism group of a certain subclass of smooth
Gizatullin surfaces with a distinguished and rigid extended divisor is
generated by automorphisms of A1-fibrations. Moreover, such surfaces provide
examples of smooth Gizatullin surfaces with a non-transitive action of the
automorphism group. Thus, they represent counterexamples to Gizatullin's
conjecture. For such surfaces we give explicit orbits of the natural action of
the automorphism group in some special cases. Further, we present their
automorphism groups as amalgamated products of two subgroups.
