An asymptotic theory of two-dimensional planetary solitary eddies is presented. Previous studies in one-and-a-half layer models have discovered special classes of radially symmetric structure which are associated with eddies of permanent form. We generalize these studies by including an active lower layer and by considering the effects of azimuthal structure. Accordingly, we stress two main results; namely, (i) permanent-form two-layer eddies with essentially arbitrary radial structure exist, provided that the eddy includes a weak imbedded dipolar asymmetry and an appropriate counter-rotating deep flow, and (ii) fluid trapped under an eddy in Taylor columns can significantly affect eddy properties if the trapped fluid possesses non-trivial potential vorticity.The structural permanency in our solutions arises from a balance between nonlinear steepening, driven by the continuity equation, and planetary dispersion. The structural asymmetries affect eddy propagation, either by dipole interaction within the layer (as occurs in modons) or by pressure forces acting between layers. The primary role of the deep counter-rotating flow is to balance the net upper-layer transport. The interesting layer-layer interaction, however, involves higher-order dynamics and is sensitive to the continuity of the potential-vorticity field. In general, these eddies trap fluid both in the upper thermocline and in the lower layer.The dominance of oceanic anticyclones over cyclones is relatively well known. A main conclusion of this study is that the class of long-lived anticyclones is considerably broader than previously realized. This may help explain the observed bias toward anticyclonic eddies. A second conclusion is that estimates of material transport by eddies may need to account for the movement of fluid outside the main bowl of the eddies.