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We present the complete structure of the N=2 super-W3 algebra, a non-linear extended conformal algebra containing the usual N=2 superconformal algebra (with generators of spins 1, 3/2, 3/2, and 2) and a higher-spin multiplet of generators with spins 2, 5/2, 5/2 and 3. We investigate various sub-algebras and related algebras, and find necessary conditions upon possibile unitary representations of the algebra. In particular, the central charge c is restricted to two discrete series, one ascending and one descending to a common accumulation point c=6. The results suggest that the algebra is realised in certain (compact or non-compact) Kazama-Suzuki coset models, including a c=9 model proposed by Bars based on SU(2,1)/U(2).
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