On a HeNe laser oscillating simultaneously at 0.63 and 3.39 µm, the effects of an axial and a transverse magnetic fields on the output as a function of a cavity detuning are investigated theoretically and experimentally. The experiment is carried out using the laser tube with Brewster windows and the resonant mirrors for 0.63 µm. The outputs are mainly characterized by 3.39 µm mode despite of low Q resonator at 3.39 µm. The output power versus cavity detuning at 0.63 µm presents the inverse pattern to the one at 3.39 µm. In the axial magnetic field, large Faraday rotation arises at 3.39 µm and it saturates in the vicinity of 60 Gauss. In the transverse magnetic field, σ modes of two wave-lengths occur when the windows have maximum transmission for σ polarization. For fields higher than about 60 Gauss, the σ mode of 3.39 µm is quenched and π mode appears because of the strong coupling of σ-π transition and its effect is enhanced by the competition between two wavelengths. The output powers versus cavity detuning agree well with our theoretical result which is analyzed by the semiclassical treatment of two transitions with a common upper level in a Zeeman laser. Furthermore the optical modulation on 3.39 µm line is also proposed by means of the competition effect.