We investigate the structure of nonlinear stationary waves propagating obliquely to the magnetic field in a cold bi-ion plasma. By using the constants of motion that follow from the multi fluid equations, the system may be described by four coupled first-order differential equations. A new constant of motion characterizing a bi-ion flow (called the ‘energy difference integral’) is found. The combination of relations between the flow speeds derived from the conservation laws, which we call the ‘momentum–energy hodographs’, reveal some important features of stationary waves and solitons. Soliton solutions representing both compressions and rarefactions in the ion fluids exist in specific windows in the ‘Alfvén Mach number–obliquity’ space. In other windows, solutions characterized by both oscillating and soliton properties (‘oscillitons’) exist. Critical Mach numbers and propagation angles narrow the size of the windows where smooth soliton solutions can be constructed.