Finite-gap solutions of self-duality equations for $SU(1,1)$ and $SU(2)$ groups and their axisymmetric stationary reductions

An extensive new class of solutions is obtained for the $SU(1,1)$ and $SU(2)$ duality equations in terms of the Riemann $\theta$-functions for a Riemann surface depending on the dynamical variables. The dynamics in the resulting solutions is thus determined by the motion of the surface in the moduli manifold. The axisymmetric stationary case is discussed, for which the solutions reduce to solutions of the vacuum Einstein equations. In the degenerate case, the class of solutions is believed to include all known solutions of the instanton and monopole type.