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Hilbert constructed an early example of a non-desarguesian plane by modifying the lines of the real plane inside an ellipse. We employ methods from elementary algebraic geometry to show that the group of automorphisms of this plane leaves the ellipse invariant (unless the ellipse is a circle). Using this result, we determine the group of automorphisms of Hilbert's plane and some generalizations.

Automorphisms of Hilbert's non-desarguesian affine plane and its projective closure