The heliospheric termination shock must exhibit asymmetry in its shape, due in part to internal latitude variations in the solar wind, and in part to the special directions defined by the external interstellar flow and/or the galactic magnetic field. This asymmetry shows up naturally in numerical simulations of the interaction between the heliosphere and local interstellar medium. However, to date only one analytical treatment of the theory has appeared (Barnes, 1998), which discussed the modification of the shock shape due to solar wind latitude variations in the presence of spherically symmetric outer boundary conditions. In the present report, we discuss an extension and generalization of this theory to an axially symmetric gasdynamic system, in which departure from spherical symmetry may be due either to internal solar wind variations or to the directional properties of the external interstellar medium. It is shown that for steady flow the post-shock region is characterized by an infinite set of quantities that are conserved along streamlines; among these invariants are the stagnation pressure and a quantity closely related to vorticity. Moreover, for a given latitude profile of the (supersonic) solar wind, the geometry of the termination shock uniquely determines the value of these invariants at the points where the streamlines emerge from the shock. In the case of flow into a static interstellar medium for which the appropriate boundary condition is that the stagnation pressure is the same for all streamlines, it can be shown that the location and shape of the termination shock, together with the entire heliosheath flow, are completely determined by the external pressure plus the internal solar wind parameters. However, for even a slightly non-static interstellar flow, or for anisotropy in exterior conditions from any other cause, such as the galactic magnetic field, exterior interstellar plus interior solar-wind conditionsdo not uniquely determinethe heliosheath flow field or the shape of the termination shock. A complete solution requires specification of additional conditions, such as the run of dynamic pressure across the distant wake, or the requirement that the heliosheath flow be vorticity-free. ©1999 American Institute of Physics.