The equatorial ionosphere contains imbedded bubbles that rise vertically through a horizontally stratified plasma. When gravity, neutral winds or external electric fields act on the F region plasma, electric fields form in the density perturbations that comprise the bubbles. The second order differential equation that describes the electric fields is a nonseparable elliptic equation. Exact solutions of this nonlinear differential equation for the electric fields and potentials are derived assuming linear or circular symmetry to the density structures imbedded in the background plasma. A wide variety of analytic solutions for electric potentials are found for both density cavities and density enhancements. These solutions may be used for quasi-analytic transport of the bubbles with equations derived from the continuity equation for the plasma. The analytic solutions may also be used to test numerical solvers of the nonseparable elliptic equations that describe the electric fields in the unstable plasma.