Assuming the four-dimensional space-time to be a general warped product of two surfaces we reduce the four-dimensional Einstein equations to a two-dimensional problem which can be solved. All global vacuum solutions are explicitly constructed and analysed. The classification of the solutions includes the Schwarzschild, the (anti-)de Sitter, and other well-known solutions but also many exact ones whose detailed global properties to our knowledge have not been discussed before. They have a natural physical interpretation describing single or several wormholes, domain walls of curvature singularities, cosmic strings, cosmic strings surrounded by domain walls, solutions with closed timelike curves, etc.
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