Finite-gap solutions of the Veselov-Novikov equation

In 1984 A.P. Veselov and S.P. Novikov obtained explicit formulas for real finite zone, two--dimensional potential Schroedinger operators. In particular5 they introduced a new integrablòe equation, known thereafter as the Veselov--Novikov equation whose real finite--gap solutions are associated to algebro-geometric data on Prym varieties. In this talk we shall review some of their results and discuss generalizations.