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This is known to be a completely integrable system, in the sense that there is an infinite sequence of formal integrals of motion $\\int h_k$, $k\\in \\mathbb{Z}_{+}$. In each $\\int h_{2k}$ the term with the highest regularity involves the Sobolev norm $\\dot H^{k}(\\mathbb{T})$ of the solution of the DNLS equation. 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