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A formalism (ζ-complex analysis), allowing one to construct global Einstein metrics by matching together local ones described in the papers [G. Sparano, G. Vilasi, A.M. Vinogradov, Differential Geom. Appl. 16 (2002) 95–120; Phys. Lett. B 513 (2001) 142–146], is developed. With this formalism the singularities of the obtained metrics are described naturally as well.
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