Azimuthal equatorial capillary-gravity flows in spherical coordinates

We work in the setting of spherical coordinates to prove existence of free-surface capillary-gravity azimuthal equatorial flows which allow for variations in the vertical direction (so currents can be accommodated) but with no meridional flow. We perform the task by deriving an implicit equation, which determines the pressure at the surface if the free surface is known, and vice-versa (given the current profile). The solutions of this equation are found by using the implicit function theorem in a proper setting, which incorporates the observed symmetry about the Equator and takes advantage of the freedom in choosing the reference for the length of the Earth's radius at the Equator.