This paper explores the integration of movable-antenna (MA) arrays into an integrated sensing and communications (ISAC) system over low-altitude platforms (LAPs) to support low-altitude economy (LAE) applications. The system involves an unmanned aerial vehicle (UAV) acting as a UAV-enabled LAP (ULAP) to provide both information transmission and sensing capabilities. To enhance throughput and meet the sensing beam pattern threshold, the paper formulates a data rate maximization problem by jointly optimizing the transmit information and sensing beamforming, as well as the antenna positions of the MA array. The non-convex nature of the problem is addressed using an efficient alternating optimization (AO)-based algorithm, which iteratively optimizes parts of the variables while fixing others. Numerical results demonstrate the superiority of the proposed MA array-based scheme in terms of achievable data rate and beamforming gain compared to fixed position and random position array schemes. The paper also discusses the robustness of the proposed scheme under varying antenna numbers and sensing thresholds, confirming its effectiveness in maximizing data rates and beamforming gains.This paper explores the integration of movable-antenna (MA) arrays into an integrated sensing and communications (ISAC) system over low-altitude platforms (LAPs) to support low-altitude economy (LAE) applications. The system involves an unmanned aerial vehicle (UAV) acting as a UAV-enabled LAP (ULAP) to provide both information transmission and sensing capabilities. To enhance throughput and meet the sensing beam pattern threshold, the paper formulates a data rate maximization problem by jointly optimizing the transmit information and sensing beamforming, as well as the antenna positions of the MA array. The non-convex nature of the problem is addressed using an efficient alternating optimization (AO)-based algorithm, which iteratively optimizes parts of the variables while fixing others. Numerical results demonstrate the superiority of the proposed MA array-based scheme in terms of achievable data rate and beamforming gain compared to fixed position and random position array schemes. The paper also discusses the robustness of the proposed scheme under varying antenna numbers and sensing thresholds, confirming its effectiveness in maximizing data rates and beamforming gains.