**Core Concept**
The atlantoaxial joint is a synovial joint that connects the first and second cervical vertebrae (C1 and C2) and is stabilized by a pair of ligaments, allowing for limited rotation and flexion. The ligaments play a crucial role in maintaining the stability of this joint.

**Why the Correct Answer is Right**
The ligament primarily responsible for maintaining atlantoaxial stability is the transverse ligament of the atlas (C1). This ligament is a strong, fibrous band that connects the posterior arch of the atlas to the anterior arch, effectively preventing the odontoid process of the axis (C2) from posteriorly displacing into the foramen magnum. The transverse ligament is also supported by the accessory ligaments, including the apical and alar ligaments.

**Why Each Wrong Option is Incorrect**
**Option B:** The anterior atlanto-occipital membrane is not primarily responsible for maintaining atlantoaxial stability; it is more relevant to the atlanto-occipital joint.

**Option C:** The posterior atlanto-occipital membrane is involved in the atlanto-occipital joint, not the atlantoaxial joint.

**Option D:** The apical ligament is an accessory ligament that provides additional support, but it is not the primary ligament responsible for maintaining atlantoaxial stability.

**Clinical Pearl / High-Yield Fact**
It is essential to remember that the transverse ligament of the atlas can be injured in cases of rheumatoid arthritis, leading to atlantoaxial instability and potential neurological complications.

**Correct Answer:** A. Transverse ligament of the atlas.