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Vertebral body compression fractures are early signs of osteoporosis. Though these fractures are visible on Computed Tomography (CT) images, they are frequently missed by radiologists in clinical settings. Prior research on automatic methods of vertebral fracture classification proves its reliable quality; however, existing methods provide hard-to-interpret outputs and sometimes fail to process cases with severe abnormalities such as highly pathological vertebrae or scoliosis. We propose a new two-step algorithm to localize the vertebral column in 3D CT images and then detect individual vertebrae and quantify fractures in 2D simultaneously. We train neural networks for both steps using a simple 6-keypoints based annotation scheme, which corresponds precisely to the current clinical recommendation. Our algorithm has no exclusion criteria, processes 3D CT in 2 seconds on a single GPU, and provides an interpretable and verifiable output. The method approaches expert-level performance and demonstrates state-of-the-art results in vertebrae 3D localization (the average error is 1 mm), vertebrae 2D detection (precision and recall are 0.99), and fracture identification (ROC AUC at the patient level is up to 0.96). Our anchor-free vertebra detection network shows excellent generalizability on a new domain by achieving ROC AUC 0.95, sensitivity 0.85, specificity 0.9 on a challenging VerSe dataset with many unseen vertebra types.