Unmanned aerial vehicle (UAV) is expected to bring transformative improvement to the integrated sensing and communication (ISAC) system. However, due to shared spectrum resources, it is challenging to achieve a critical trade-off between these two integrated functionalities. To address this issue, we propose in this paper a new integrated periodic sensing and communication mechanism for the UAV-enable ISAC system. Specifically, the user achievable rate is maximized via jointly optimizing UAV trajectory, transmit precoder, and sensing start instant, subject to the sensing frequency and beam pattern gain constraints. Despite that this problem is highly non-convex and involves an infinite number of variables, we obtain the optimal transmit precoder and derive the optimal achievable rate in closed-form for any given UAV location to facilitate the UAV trajectory design. Furthermore, we first prove the structural symmetry between optimal solutions in different ISAC frames without location constraints and then propose a high-quality UAV trajectory and sensing optimization algorithm for the general location-constrained case. Simulation results corroborate the effectiveness of the proposed design and also unveil a more flexible trade-off in ISAC systems over benchmark schemes.