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An aim of magnetic resonance electrical impedance tomography (MREIT) is to visualize the internal current density and/or the conductivity of an imaging object. In MREIT, it is desirable to use just one component of the internal magnetic flux density vector B=(B x,B y,B z) caused by the injected current, measured without rotating the object. We present a method of visualizing the axial anisotropic conductivity tensor by use of the measured magnetic flux density B z data. The method involves the use of a projected current density, which is a uniquely and stably determined component of the internal current generated by the injected current, derived from the measured B z data. Each component of the axial anisotropic conductivity is recovered by matching the measured B z data with a determined intermediate isotropic conductivity and the projected currents. Results from numerical simulations demonstrate that the proposed algorithm is robust to noise and stably determines the anisotropic conductivity tensor on the imaging slice. For a practical implementation, we studied a postmortem canine brain case to visualize each component of the anisotropic conductivity. We observed that the reconstructed anisotropic conductivity images clearly reflects the anisotropic property of the white matter in the direction parallel to its fibers.