Geodesic
Theta Function Solutions of the Schlesinger System and the Ernst Equation
Funkcionalʹnyj analiz i ego priloženiâ, Tome 34 (2000) no. 4, pp. 18-34

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We establish a link between the Schlesinger system and the Ernst equation (the stationary axisymmetric Einstein equation) on the level of algebro-geometric solutions. We calculate all metric coefficients corresponding to general algebro-geometric solutions of the Ernst equation. 
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D. A. Korotkin; V. B. Matveev. Theta Function Solutions of the Schlesinger System and the Ernst Equation. Funkcionalʹnyj analiz i ego priloženiâ, Tome 34 (2000) no. 4, pp. 18-34. http://geodesic.mathdoc.fr/item/FAA_2000_34_4_a1/