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Abstract: A CR twistor space is the CR manifold canonically associated with a conformal 3-manifold. In the flat case, it is realized by the split hyperquadric Q^{2,2} in complex projective 3-space. We study the flat CR twistor model Q^{2,2} in complex projective 3-space by explicit projective methods. Using the anti-holomorphic involution j induced by the twistor fibration, we classify the projective lines contained in Q{2,2} into twistor fibres and transverse lines, and relate the latter to round 2-spheres in S3 through an explicit incidence correspondence. We then analyze hyperplane sections and smooth j-invariant quadric sections under the j-compatible symmetry group. In the disjoint case, the construction yields an explicit one-parameter family of globally defined real-analytic Levi-nondegenerate non-spherical CR structures on S^3.