  Vibrations of an elastic rod are described by a Sturm-Liouville system. We
present a general discussion of isospectral (spectrum preserving) deformations
of such a system. We interpret one family of such deformations in terms of a
two-component modified Camassa-Holm equation (2-mCH) and solve completely its
dynamics for the case of discrete measures (multipeakons). We show that the
underlying system is Hamiltonian and prove its Liouville integrability. The
present paper generalizes our previous work on interlacing multipeakons of the
2-mCH and multipeakons of the 1-mCH. We give a unified approach to both
equations, emphasizing certain natural family of interpolation problems germane
to the solution of the inverse problem for 2-mCH as well as to this type of a
Sturm-Liouville system with singular coefficients.
