Ernst equation, Fay identities and variational formulas on hyperelliptic   curves
  We present a unified approach to theta-functional solutions of the stationary axisymmetric Einstein equations in vacuum. Using Fay's trisecant identity and variational formulas on hyperelliptic Riemann surfaces, we establish formulas for the metric functions, the Ernst potential and their derivatives. 
C.Klein, D.Korotkin, V.Shramchenko