What is the primary requirement for a satellite to achieve a Geostationary Orbit ($e=0$)?

Its orbit must be perfectly circular ($e=0$) and located above the equator

A Geostationary Orbit (GEO) is a highly specific, practical application of orbital mechanics. For a satellite to remain fixed above the same point on the Earth's surface, two conditions must be rigorously met. First, the orbital period must match the Earth's rotation period, which dictates a specific radius. Second, the orbit must be perfectly circular, meaning its eccentricity must be exactly zero ($e=0$). If the orbit possessed any eccentricity, the satellite would appear to oscillate north and south relative to the ground observer throughout its orbital period; this movement is described as a ground track oscillation. Therefore, ground controllers must apply thruster burns to continuously reduce any measured eccentricity to zero to maintain the required fixed position.