Extending symmetric designs

If a symmetric 2-design with parameters (v, k, λ) is extendable, then one of the following holds: v = 4λ + 3, k = 2λ + 1; or v = (λ + 2)(λ² + 4λ + 2), k = λ² + 3λ + 1; or v = 111, k = 11, λ = 1; or v = 495, k = 39, λ = 3. In particular, there are at most three sets of extendable symmetric design parameters with any given value of λ. As a consequence, the only twice-extendable symmetric design is the 21-point projective plane.