The propagation of a superluminous plane wave of fixed profile through a cold electron plasma, in a direction perpendicular to the ambient magnetostatic field, is considered. The polarization is such that the electric vector has no component parallel to the magnetostatic field. The analysis is referred to the frame of reference in which all the field quantities are uniform in space, and the exact governing equations are cast into the form of an autonomous system of four nonlinear, first-order, coupled differential equations. The equations contain the parameter n that specifies the wave speed, c/n, in the laboratory frame, and inspection reveals that the limiting case n = 0 admits two exact solutions, which are monochromatic, of arbitrary amplitude, and recover the standard solutions of linear theory in the small-amplitude limit. The possibility of the existence of corresponding period solutions for the case n ≫ 1 is then investigated by a perturbation treatment. I t is found that the analysis can be carried through explicitly, and, to the first order in n, the outcome confirms the existence of periodic solutions, and yields the dependence of the period'on an amplitude parameter. The results are checked against a direct numerical integration procedure that incorporates a method of searching for periodic solutions, and close quantitative agreement is found for sufficiently small values of n.