The exact theory of plane waves of fixed profile travelling with constant velocity cZ/n (0 ≤ n ≤ 1) through a uniform, cold, electron-ion plasma in a magneto-static field is examined in terms of the governing equations referred to the frame of reference in which there is no space dependence. Canonical periodic solutions are defined as those with zero average rate of flow of electrons (and a fortiori of ions) in the laboratory frame. It is shown that the equations lead to a second- order, nonlinear, ordinary vector differential equation for the reduced velocity u of the electrons. A scalar first integral is obtained, from which it is deduced that the path in u-space of any solution lies within a bounded domain. It is shown that, for propagation across the magnetic field, a polarization is possible in which the particle velocities and the electric field are orthogonal to the magnetic field. The simpler model of an electron Plasma is considered. Explicit canonical periodic solutions, with the stated polarization, are obtained for propagation across the magnetic field in the case n = 0 and the case n ≃ 1. These support the conjecture that, for any fixed value of n in [0, 1], there are two ‘modes’ of arbitrary amplitude which reduce to the familiar monochromatic waves of linear magneto-ionic theory in the small amplitude limit.