  We solve the Cauchy problems for the $\mu$-Camassa-Holm integro-partial
differential equation of Khesin-Lenells-Misio\l{}ek and its non-quasilinear
version introduced by Qu-Fu-Liu in the complex-analytic framework. These
equations have nonlocal nature at two levels: they involve a
pseudo-differential operator of negative order and this operator is defined in
terms of an integral of the unknown function. We prove unique solvability of
the Cauchy problems and provide an estimate of the lifespan of the solutions.
Our method is the Ovsyannikov type argument by Batrostichi-Himonas-Petronilho
about a scale of Banach spaces of analytic functions, but the
non-quasilinearity is dealt with in a different way from theirs. Indeed, we use
a simple reduction which is just a nonlocal version of a classical trick.
