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The Brillouin flow equilibrium in magnetically insulated lines is a state in which field-emitted electrons, confined to a sheath near the cathode by a magnetic field, drift laminarly at the Â¿Â¿ x Â¿Â¿ velocity in the self-consistent fields of the anode-cathode gap. Herein, this state is perturbed by a TM disturbance traveling along the direction of electron flow. Asymptotic expansions of all quantities of interest in some small parameter are performed in the long wavelength limit. First order quantities are seen to be governed by the Korteweg-deVries' equation, which admits soliton solutions. In the appropriate limits of either infinite wavelength or vanishingly small amplitude, the results contained herein are seen to agree with other analyses, based, respectively, upon either the nondispersive telegraphers' equations or a linearized analysis.