Summary: The existence of multiple solutions for a shrinking surface flow was first reported in the middle of the last decade. Since then, the exploration of second or even a third solution for the shrinking surface flows is a topic of great interest. Consequently, a huge volume of literature has been produced on this topic during the last one and half decade. According to such an available literature a sufficient amount of wall suction is mandatory for the existence of dual solutions. However, in the current study, it is shown that for an unsteady shrinking cylinder the existence of a second solution is not conditional to the provision of sufficient wall suction; rather it is possible in the case of no wall suction or even in the presence of sufficiently weak wall injection, also. Such a happening of the duality of solution for a shrinking surface flow in the absence of wall suction or even in the presence of wall injection is a novel finding. To the best of authors’ knowledge such kind of results have never been reported in the past. In this regard, the transverse surface curvature of the cylinder’s surface is observed to play an important role.