On inversion of the Abel and Abel-Prym maps

Abelian variety called Jacobian of the Riemann surface. Finding out the full preimage of a point under the Abel map is the content of the Jacobi inversion problem. That is a classical problem resolved by Riemann. It also has many applications especially in the theory of integrable systems, based on the fact that Jacobians often play the role of invariant tori. However, for majority of classical and new integrable systems Lagrangian tori are not Jacobians but different Abelian varieties called Prym varieties, or Prymians. I shall describe peculiarities of the Jacobi inversion problem on Prymians, including a new case of its solvability, with a special attention to the computational aspect.