  We are exploring variations of the Novikov equation that have weak solutions
called peakons. Our focus is on a two-component Novikov equation with a
non-self-adjoint $4\times 4$ Lax operator for which we examine the related
forward and inverse spectral maps for the peakon sector. To tackle the forward
spectral problem, we convert it into a matrix eigenvalue problem, for which the
original boundary value problem is a two-fold cover. We then use an isospectral
deformation to the long-time regime to calculate the relevant eigenvalues. To
support the long-term deformation, we prove the global existence of the peakon
flows using ideas based on Moser's deformation method, which was used in his
study of the finite Toda lattice. We subsequently solve the inverse problem by
studying a trio of Weyl functions that can be approximated simultaneously by
rational functions that involve tensor products and an additional symmetry
condition. This part of our paper is rooted in Krein's solution to the inverse
problem for the Stieltjes string.
