The ITCZ lies at the ascending branch of the tropical meridional overturning circulation, where nearsurface meridional mass fluxes vanish. Near the ITCZ, column-integrated energy fluxes vanish, forming an atmospheric energy flux equator (EFE). This paper extends existing approximations relating the ITCZ position and EFE to the atmospheric energy budget by allowing for zonal variations. The resulting relations are tested using reanalysis data for 1979-2014. The zonally varying EFE is found as the latitude where the meridional component of the divergent atmospheric energy transport (AET) vanishes. A Taylor expansion of the AET around the equator relates the ITCZ position to derivatives of the AET. To a first order, the ITCZ position is proportional to the divergent AET across the equator; it is inversely proportional to the local atmospheric net energy input (NEI) that consists of the net energy fluxes at the surface, at the top of the atmosphere, and zonally across longitudes. The first-order approximation captures the seasonal migrations of the ITCZ in the African, Asian, and Atlantic sectors. In the eastern Pacific, a third-order approximation captures the bifurcation from single-to double-ITCZ states that occurs during boreal spring. In contrast to linear EFE theory, during boreal winter in the eastern Pacific, northward cross-equatorial AET goes along with an ITCZ north of the equator. EFE and ITCZ variations driven by ENSO are characterized by an equatorward (poleward) shift in the Pacific during El Niño (La Niña) episodes, which are associated with variations in equatorial ocean energy uptake.