  Consideration here is a generalized $\mu$-type integrable equation, which can
be regarded as a generalization to both the $\mu$-Camassa-Holm and modified
$\mu$-Camassa-Holm equations. It is shown that the proposed equation is
formally integrable with the Lax-pair and the bi-Hamiltonian structure and its
scale limit is an integrable model of hydrodynamical systems describing short
capillary-gravity waves. Local well-posedness of the Cauchy problem in the
suitable Sobolev space is established by the viscosity method. Existence of
peaked traveling-wave solutions and formation of singularities of solutions for
the equation are investigated. It is found that the equation admits a single
peaked soliton and multi-peakon solutions. The effects of varying
$\mu$-Camassa-Holm and modified $\mu$-Camassa-Holm nonlocal nonlinearities on
blow-up criteria and wave breaking are illustrated in detail. Our analysis
relies on the method of characteristics and conserved quantities and is
proceeded with a priori differential estimates.
