Solutions of two-dimensional einstein equations parametrized by arbitrary functions and generated by the O(2, 1) $\sigma$ model

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.