- This paper presents a perturbation approach for determining relative motion initial conditions for periodic motion in the vicinity of a Keplerian elliptic orbit of arbitrary eccentricity, as well as an analytical solution for the relative orbit that accounts for quadratic nonlinearities in the differential gravitational acceleration. The analytical solution is obtained in the phase space of the rotating coordinate system, centered at the reference satellite, and is developed in terms of a small parameter relating relative orbit size, and semi-major axis and eccentricity of the reference orbit. The results derived are applicable for arbitrary epoch of the reference satellite. Relative orbits generated using the methodology of this paper remain bounded over much longer periods in comparison to the results obtained using other approximations found in the literature, since the semi-major axes of the satellites are shown to be matched to the second order in the small parameter. The derived expressions thus serve as excellent guesses for initiating a numerical procedure for matching the semi-major axes of the two satellites. Several examples support the claims in this paper.