Integrated sensing and communication (ISAC) technologies have garnered substantial academic and industrial interest. ISAC is recognized for its ability to leverage unified infrastructure and waveforms to simultaneously transmit information and receive echoes, thereby significantly enhancing the efficiency of spectrum, cost, and energy. Unmanned aerial vehicles (UAVs) offer flexible observation and enhanced communication capabilities, poised to revolutionize ISAC systems. Existing ISAC research often overlooks practical asymmetric sensing and communication needs, leading to suboptimal resource utilization. To address this, we propose an integrated periodic sensing and communication mechanism for UAV-enabled ISAC. Our approach optimizes UAV trajectory, user association, sensing selection, and beamforming to maximize system achievable rates while meeting specific sensing and beam pattern requirements. We derive closed-form solutions for beamforming and propose efficient algorithms to solve the non-convex optimization challenges, demonstrating significant improvements over traditional approaches.