Integrable isotropic profiles for polarized light

We consider the propagation of polarized light in the medium with an isotropic refraction index profile and show that polarization violates the additional symmetries of the medium. Then we suggest a scheme for the construction of polarization-dependent refraction index which restores all symmetries of the initial profile. We illustrate the proposed scheme on the examples of Luneburg and Maxwell's fisheye profiles.Comment: Based on talk given by M.D XVIII Conference "Symmetry Methods in Physics", 10-16 July, 2022, Yerevan. Submitted to proceedings. arXiv admin note: substantial text overlap with arXiv:2106.1546