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The equations to be considered are $\\mu$CH of Khesin-Lenells-Misio\\l{}ek, $\\mu$DP of Lenells-Misio\\l{}ek-Ti\\u{g}lay, the higher-order $\\mu$CH of Wang-Li-Qiao and the non-quasilinear version of Qu-Fu-Liu. We prove the unique local solvability of the Cauchy problems and provide an estimate of the lifespan of the solutions. Moreover, we show the existence of a unique global-in-time analytic solution for $\\mu$CH, $\\mu$DP and the higher-order $\\mu$CH. 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