An extended Abel–Jacobi map

We solve the problem of inversion of an extended Abel–Jacobi map ∫P0P1ω+⋯+∫P0Pg+n−1ω=z,∫P0P1Ωj1+⋯+∫P0Pg+n−1Ωj1=Zj,j=2,…,n, where Ωj1Ωj1 are (normalized) Abelian differentials of the third kind. In contrast to the extensions already studied, this one contains meromorphic differentials having a common pole Q1Q1. This inversion problem arises in algebraic geometric description of monopoles, as well as in the linearization of integrable systems on finite-dimensional unreduced coadjoint orbits on loop algebras.