Prym varieties of branched coverings and nonlinear equations

An efficacious realization is presented of finite-gap solutions of the Veselov–Novikov equation expressed in terms of the theta function of Prym varieties of double coverings of algebraic curves with two branch points. For the given Prym mapping equations are obtained which locally solve a problem of Riemann–Schottky type, and a local Torelli theorem is proved.