<p>We prove the existence of solutions for the axisymmetric steady Navier–Stokes system around an infinite cylinder under external forces. The solutions are constructed to be decaying at the horizontal infinity, despite an analogue of the Stokes paradox for the linearized system, and having neither periodicity nor decay in the vertical direction. The proof is based on perturbation of the nonlinear system around a suction flow. The class of functions in this paper, which is a subspace of the space of Fourier transformed vector finite Radon measures, is inspired by Giga-Saal (2013) treating rotating boundary layers.</p>