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We consider the Lie algebra that corresponds to the Lie pseudogroup of all conformal transformations on the plane. This conformal Lie algebra is canonically represented as the Lie algebra of holomorphic vector fields in ℝ2≃ℂ. We describe all representations of    via vector fields in J 0ℝ2=ℝ3(x,y,u), which project to the canonical representation, and find their algebra of scalar differential invariants.

Differential Invariants of the 2D Conformal Lie Algebra Action