We solve the problem of inversion of an extended Abel–Jacobi map ∫ P 0 P 1 ω + ⋯ + ∫ P 0 P g + n − 1 ω = z , ∫ P 0 P 1 Ω j 1 + ⋯ + ∫ P 0 P g + n − 1 Ω j 1 = Z j , j = 2 , … , n , where Ω j 1 are (normalized) Abelian differentials of the third kind. In contrast to the extensions already studied, this one contains meromorphic differentials having a common pole Q 1 . 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.