Rotationally symmetric deformations for which both principal stresses are nonnegative everywhere in a flat annular membrane subjected to an axial surface load, are treated. The mathematical formulation leads to some nonlinear two-point boundary value problem (b.v.p.) for the axisymmetric deformations with also nonlinear boundary condition. For this problem the author considers wrinkle-free solutions, which leads to nonnegative deformations satisfying an also nonnegative condition involving the solution and its first derivative. The second condition is in connection with the circumferential stress. Using the concept of superfunctions of the b.v.p. in relation with concavity and monotonicity arguments the author proves the existence of some special solutions to the b.v.p. A maximum principle for required solutions is proved under certain natural hypotheses which reduce the nonnegativity of the circumferential stress to the nonnegativity of the values of some appropriate functions at the ends of the interval. The author introduces the concept of switch point at which the stress circumferential components vanish at both inner and outer edges of the annulus, and proves the uniqueness and existence of such switch point solutions. The connection between the parametric domain of wrinkle-free solutions and switch points is given explicitly. Numerical investigation using the shooting method is performed. Wrinkle-free solutions are also treated involving displacement data, and the parameter domain of such solutions is determined.