  We solve Cauchy problems for some $\mu$-Camassa-Holm integro-partial
differential equations in the analytic category. 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. The present work is
the first result of such a global nature for these equations.
  AMS subject classification: 35R09, 35A01, 35A10, 35G25
