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The space ${\\cal D}^k\\_{\\lambda,\\mu}(S^1)$ of $k$-th order linear differential operators from ${\\cal F}\\_\\lambda(S^1)$ to ${\\cal F}\\_\\mu(S^1)$ is a natural module over $\\mathrm{Diff}(S^1)$, the diffeomorphism group of $S^1$. We determine the algebra of symmetries of the modules ${\\cal D}^k\\_{\\lambda,\\mu}(S^1)$, i.e., the linear maps on ${\\cal D}^k\\_{\\lambda,\\mu}(S^1)$ commuting with the $\\mathrm{Diff}(S^1)$-action. 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