The aim of this paper is to analyze the propagation of large-amplitude Alfvén waves in a perfectly conducting plasma in the magnetohydrodynamic approximation, for cylindrical magnetic tubes. In a similar way to that done in Sallago and Platzeck, instead of linearizing the magnetohydrodynamic equations and searching monochromatic waves, we impose the conditions that the perturbations propagate without distortion with a group velocity parallel to the background magnetic induction field, in the reference system in which the plasma is locally at rest, the perturbations are incompressible, there exists a relation between the velocity and induction magnetic field perturbations, and the total pressure remains constant. The solutions are similar to the case in which the background fields vary with a rectilinear variable. When the solution is searched using this methodology there is no singularity problem. If the background total pressure is not uniform and constant, it is not possible to find waves of large amplitude that propagate like Alfvén waves in cylindrical magnetic tubes.