Exact ray trajectories in anisotropic media with axially varying non-diagonal optical metrics

In this study, closed-form ray trajectories are derived for several classes of anisotropic inhomogeneous media with variation along one Cartesian axis. The derivation starts from conserved transverse momentum and expresses the ray integral through the Schur complement of a real symmetric optical metric for a mode with quadratic dispersion. Evaluation of this integral produces explicit trajectories for reciprocal and quadratic scale profiles with a tilted optical axis, a linear scale profile in a uniaxial hyperbolic medium, heliconical rotation at zero transverse momentum, and axial coordinate shear. The metric may contain all three off-diagonal entries, and explicit conditions specify where the trajectories remain real with finite slopes. Ordinary and extraordinary trajectories can be compared explicitly for the three uniaxial examples that support both modes. For each of the five closed-form solutions, ray positions calculated for three parameter values agree with those obtained by independent integration of the geodesic equations and by quadrature of the ray integral.