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.
Categories: math-ph math.AG math.MP nlin.SI