We have derived the corresponding equations and found their solutions both for nonparaxial and paraxial beams. The paraxial solutions we have presented in the form of the generalized Hermite-Gaussian beams propagating perpendicular to the optical axi
s of a uniaxial crystal. We have also constructed the generalized Laguerre-Gaussian beams at the z=0 plane and analyzed their evolution in a homogeneous isotropic medium. Comparing it with the evolution of the standard Laguerre-Gaussian beams with and in the crystal we have revealed that the additional elliptic deformation of the extraordinary beam results in topological reactions that essentially distorts field structure for the account of different rotation rates of the vortex row originated from the centered degenerate optical vortex and the conoscopic pattern. We have predicted conversion of the vortex topological charge at the beam axis similar to that in astigmatic lenses and analyzed the radical differences with this process. We have revealed the synchronic oscillations of the spin angular momentum and the sign of the vortex topological charge at the beam axis.
