We develop a fully nonlinear theory of electrostatic stationary waves of the ion-cyclotron-acoustic type propagating in a magnetized plasma. The analysis is facilitated by the existence of two fundamental constants of the motion, namely conservation of momentum flux parallel to the ambient field, and conservation of energy in the direction of propagation of the wave. These lead to the structure of the waves being governed by a first-order differential equation for the longitudinal ion flow speed and, therefore, its properties are readily analysed. In the case of subsonic wave speeds, compressive soliton wave structures propagate in a cone of obliquity around the magnetic field. If the wave speed is supersonic, the waves are oscillatory in nature, and asymmetric about the compressive and rarefactive phases, essentially because, in the compressive phase, the flow is driven towards its sonic point, whereas in the rarefactive phase, the flow is driven away from its sonic point. As the initial ‘driver’ field approaches a critical value, the end point of the compression coincides with the sonic point, with the result that the waveform in the flow speed (and potential) forms a cusp. It is possible that this feature may provide an explanation for the spiky waveforms observed by FAST and other similar satellite observations.