A non-coplanar flyby is recognized as an effective method for monitoring high-value target satellites in geosynchronous orbit. Taking the scenario of patrolling the geosynchronous belt with an elliptical orbit as an example, a method for solving the minimal orbital intersection distance between any two elliptic orbits was proposed, and it was proven that the optimal flyby point for the patrolling satellite is the ascending node. ATK（aerospace tool kit）was used to model the mission scenario, investigating the variations in the optimal flyby point position in J2 gravitational model. The variation law in terminal constraints (e.g., relative distance and sun phase angle at the flyby point) were studied under different local times and flyby directions. Aoptimal fuel transfer strategy satisfying the maximum transfer time and terminal constraints was proposed, and mission parameters for the mission under different gravitational model assumptions were solved by using the maneuver planning module in ATK. Mission parameters corresponding to different true anomaly angles in the two-body model were analyzed, the effectiveness of the strategy was validated, and a basis for planning multiple patrol sequences was provided.