This article examines the cache-enabled high-low-altitude-platforms integrated network (CHLIN), which consists of multiple high-altitude platforms (HAPs) and cacheable low-altitude platforms (LAPs). CHLIN aims to leverage the edge caching, the flexibility of LAPs and the broad coverage and stability of HAPs to realize multi-ground-user content transmission. Considering the low endurance, dynamics, and limited storage capacity of LAPs, a combined optimization of content caching policies, offloading decisions, and HAP-servers and LAP-servers selection is designed to reduce the delay of content transmission while fulfilling users' demand for the quality of service. We transform the complex non-convex optimization problem with highly coupled variables into an equivalent convex problem. Afterward, a genetic-algorithm-embedded distributed alternating direction method of multipliers (GA-DADMM) is proposed, which adopts a distributed architecture for alternating iteration and introduces a genetic algorithm to derive the multi-dimensional and coupled local variables. Simulation results show that GA-DADMM achieves better convergence than the comparison algorithm, which is proper for large-scale optimization problems. The superiority of the proposed edge caching scheme in transmission delay reduction is also validated.