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Then we show that an orthogonal almost complex structure $J_f$ on $S^{2n}$ is integrable if and only if the corresponding section $f\\colon\\; S^{2n}\\to \\widetilde{\\cal J}(S^{2n}) $ is holomorphic. These shows there is no integrable orthogonal complex structure on the sphere $S^{2n}$ for $n>1$. We also show that there is no complex structure in a neighborhood of the space $\\widetilde{\\cal J}(S^{2n})$. 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We show that the twistor space $\\widetilde{\\cal J}(S^{2n})$ is a Kaehler manifold. Then we show that an orthogonal almost complex structure $J_f$ on $S^{2n}$ is integrable if and only if the corresponding section $f\\colon\\; S^{2n}\\to \\widetilde{\\cal J}(S^{2n}) $ is holomorphic. These shows there is no integrable orthogonal complex structure on the sphere $S^{2n}$ for $n>1$. We also show that there is no complex structure in a neighborhood of the space $\\widetilde{\\cal J}(S^{2n})$. 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