Geodesic
Solutions of two-dimensional einstein equations parametrized by arbitrary functions and generated by the O(2, 1) $\sigma$ model
Teoretičeskaâ i matematičeskaâ fizika, Tome 64 (1985) no. 1, pp. 51-60

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Solutions, parametrized by arbitrary functions, of the two-dimensional Einstein equations are constructed by making a special choice of the determinant of the metric and using solutions of the attxiliary $O(2,1)$ $\sigma$ model. These solutions generalize ones considered earlier [1]. The construction is also applied to the Einstein–Maxwell equations and to the case when matter with equation of state $\varepsilon=p$ is present. 
@article{TMF_1985_64_1_a5,
     author = {M. G. Tseitlin},
     title = {Solutions of two-dimensional einstein equations parametrized by arbitrary functions and generated by the {O(2,} 1) $\sigma$ model},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {51--60},
     publisher = {mathdoc},
     volume = {64},
     number = {1},
     year = {1985},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1985_64_1_a5/}
}
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M. G. Tseitlin. Solutions of two-dimensional einstein equations parametrized by arbitrary functions and generated by the O(2, 1) $\sigma$ model. Teoretičeskaâ i matematičeskaâ fizika, Tome 64 (1985) no. 1, pp. 51-60. http://geodesic.mathdoc.fr/item/TMF_1985_64_1_a5/