  We solve the problem of inversion of an extended Abel-Jacobi map $$
\int_{P_{0}}^{P_{1}}\omega +...+\int_{P_{0}}^{P_{g+n-1}}\omega ={\bf z}, \qquad
\int_{P_{0}}^{P_{1}}\Omega_{j1}+... +\int_{P_{0}}^{P_{g+n-1}}\Omega_{j1}
=Z_{j},\quad j=2,...,n, $$ where $\Omega_{j1}$ are (normalised) 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.
