Finite-gap solutions of the stationary axisymmetric Einstein equation in vacuum

A new and large class of exact solutions of the stationary axisymmetric Einstein equation, which are expressed in terms of the Riemann $\theta$ function, is constructed. The properties of the constructed “finite-gap” solutions differ significantly from those of the well-known finite-gap solutions (for example, of the Korteweg–de Vries equation and the nonlinear Schrödinger equation). In particular, the dependence on the dynamical variables in the final expressions is given by a trajectory on a manifold of moduli of algebraic curves, and not on the Jacobi manifold of a given curve. In a degenerate case the constructed solutions include all the main known solutions that can be expressed in terms of elementary functions.