We investigate the structure of nonlinear stationary waves propagating transverse and obliquely to the magnetic field in a cold plasma consisting of two ion populations, protons and alpha particles. By using the constants of motion which follow from the multifluid equations, the system may be described by a single first-order differential equation for the transverse case and four coupled first-order differential equations in the case of oblique propagation. In the transverse case solitons exist if the wave speed lies between the Alfven speed, based on the total mass density, and the Alfven speed modified by the density and charge ratios. At speeds in excess of this latter velocity, periodic solutions exist in which protons and alphas gyrate around each other. An analog of Rankine-Hugoniot type relations for the amplitude of the solitons and periodic waves is found for structures propagating transverse to the magnetic field. In the case of an oblique stationary wave it is shown that the tip of the proton (alpha ion) flow velocity vector moves on the surface of a sphere whose radius is determined only by the obliquity and the wave speed. Soliton solutions representing both compressions and rarefactions in the ion fluids exist in specific windows in the Alfven Mach number-obliquity space. In other windows, solutions characterized by both oscillating and soliton properties (oscillitons) exist. Critical Mach numbers and critical propagation angles limit the size of the windows in which smooth solitons can be constructed.