What is the electromagnetic field due to an alternating current in an anisotropic medium?

We know that an infinite, alternating, straight line current generates an outward cylindrically propagating electromagnetic field with the axial electric field expressed as a Hankel function of the radial distance from the current source and the azimuthal magnetic field expressed as the radial derivative of the Hankel function. All other components of the electromagnetic field are vanishing and there is no dependence on the azimuthal or axial coordinates owing to the azimuthal and axial symmetry.

If we embed such a source in an infinite, homogeneous, isotropic material, the above statements still hold. My question concerns what happens when the material is anisotropic (but remains homogeneous). For simplicity, I'm interested in a non-magnetic, biaxial, dielectric material where one of the principal axes associated with the permittivity is parallel to the line current.

I'm not clear whether I can even assume the electric field remains axial. I'm certain that the magnetic field will no longer be azimuthal as we no longer have the azimuthal symmetry, which leads me at a loss with how to proceed. I suspect the solution may involve an elliptical propagating field, possibly described by Mathieu functions, but not knowing which components of the fields are non-vanishing means I'm struggling to get going with the problem.

Thanks in advance for any help.