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A spectral and the inverse spectral problem are studied for the two-componentmodified Camassa-Holm type for measures associated to interlacing peaks. It isshown that the spectral problem is equivalent to an inhomogenous string problemwith Dirichlet/Neumann boundary conditions. The inverse problem is solved byStieltjes' continued fraction expansion, leading to an explicit construction ofpeakon solutions. Sufficient conditions for the global existence in $t$ aregiven. The large time asymptotics reveals that, asymptotically, peakons pair upto form bound states moving with constant speeds. The peakon flow is shown toproject to one of the isospectral flows of the finite Kac--van Moerbekelattice.

Multipeakons of a two-component modified Camassa-Holm equation and the relation with the finite Kac-van Moerbeke lattice