This work presents alternative formulations of the asymptotic, inhomogeneous wave tracking theory (IWT) proposed by Choudhary and Felsen and investigates the basic analytical features of the ensuing operators. The outstanding deviations from classical ray theory are traceable to the fact that the generalized system of equations is of the quasi-linear, elliptical type. Consequently, representation via an initial value formulation, standardly posed by geometrical optics (and in the quasi-homogeneous limit, by IWT) must be approached with caution, and an appropriate boundary value formulation should be considered. Questions of existence, uniqueness, stability, and locality are discussed. The following alternative, but by no means equivalent, formulations are considered: (1) first-order, quasi-linear system of equations cast into a canonical form, the existence and uniqueness (but not the stability) of whose solution is ensured by the Cauchy-Kowalewsky conditions, (2) a quasi-linear, second-order, elliptical differential equation subject to Dirichlet boundary conditions and to the requirements of elliptical uniformity, (3) same as formulation 2, but posed as a two-dimensional variational problem, and (4) quasi-linear, second-order, phase trajectory equation obtained from formulation 2 via the so-called hodograph transformation.