This paper investigates a movable-antenna (MA) array empowered integrated sensing and communications (ISAC) over low-altitude platform (LAP) system to support low-altitude economy (LAE) applications. In the considered system, an unmanned aerial vehicle (UAV) is dispatched to hover in the air, working as the UAV-enabled LAP (ULAP) to provide information transmission and sensing simultaneously for LAE applications. To improve the throughput capacity, we formulate a data rate maximization problem by jointly optimizing the transmit information and sensing beamforming and the antenna positions of the MA array. Since the data rate maximization problem is non-convex with highly coupled variables, we propose an efficient alternation optimization based algorithm, which iteratively optimizes parts of the variables while fixing others. Numerical results show the superiority of the proposed MA array-based scheme in terms of the achievable data rate and beamforming gain compared with two benchmark schemes.