  The isospectral deformations of the Frobenius-Stickelberger-Thiele (FST)
polynomials introduced in [32](Spiridonov et al. Commun. Math. Phys.
272:139--165, 2007 ) are studied. For a specific choice of the deformation of
the spectral measure, one is led to an integrable lattice (FST lattice), which
is indeed an isospectral flow connected with a generalized eigenvalue problem.
In the second part of the paper the spectral problem used previously in the
study of the modified Camassa-Holm (mCH) peakon lattice is interpreted in terms
of the FST polynomials together with the associated FST polynomials, resulting
in a map from the mCH peakon lattice to a negative flow of the finite FST
lattice. Furthermore, it is pointed out that the degenerate case of the finite
FST lattice unexpectedly maps to the interlacing peakon ODE system associated
with the two-component mCH equation studied in [17](Chang et al. Adv. Math.
299:1--35, 2016).
