The authors consider the two-point boundary-value problem resulting from the equations of nonlinear elastostatics for azimuthal shear of a Blatz-Ko tube. Previous work on this problem by Simmonds and Warne includes a numerical study of these equations and indicates that smooth radial deformation solutions (no kinks) should exist regardless of the aspect ratio of the tube, provided that the dimensionless applied torque r is small enough (r <-0.72). The numerics of Simmonds and Warne also indicated that the existence of smooth solutions for r >-0.72 depends on the geometry of the tube, and that for r = A, no smooth solution exists. Motivated by this numerical work, the authors prove via a topological shooting argument the existence and uniqueness of smooth solutions to this problem for r < tr, = 3/44 S 0.69, and the nonexistence of smooth solutions for r = a.