  This work deals with a special case of family of birational maps f : C2 -> C2
dynamically classified in [9]. In this work we study the zero entropy sub
families of f. The sequence of degrees dn associated to the iterates of f is
found to grow periodically, linearly, quadratically or exponentially. Explicit
invariant fibrations for zero entropy families and all the integrable and
periodic mappings inside the family f are given.
