Nonlinear constant‐profile Alfve´n waves are studied in an ionized gas in which thermal motions are negligible. The system of differential equations possesses numerous integrals, and can be solved up to a single quadrature. Many properties of the waves can be inferred without performing any numerical integrations. The waves necessarily involve nonzero magnetic field components in both directions perpendicular to the direction of propagation, and plane polarized waves are impossible except in the special case of equal mass particles. Circularly polarized modes are also a special case, and, unlike the more general solutions, involve neither longitudinal electric fields nor compression of the gas.