A method is given for investigating the behaviour of electromagnetic fields in a homogeneous, anisotropic medium characterized by a dielectric tensor which is diagonal, with two of the three diagonal elements equal. It is shown that each such field is related by a simple scaling procedure to a corresponding vacuum field. The vacuum field is expressed as the superposition of a transverse magnetic field, in which the magnetic vector is everywhere perpendicular to the axis of symmetry of the anisotropic medium, and a coplanar transverse electric field; and different scaling is applied separately to each partial field. The method is illustrated by application to plane-wave and dipole fields. In particular, the field is obtained of a time-harmonic electric dipole of arbitrary orientation situated in the anisotropic medium.