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We present a characterization of $2$-dimensional Lorentzian manifolds withconstant Ricci scalar curvature. It is well known that every $2$-dimensionalLorentzian manifolds is conformally flat, so we rewrite the Ricci scalarcurvature in terms of the conformal factor and we study the solutions of thecorresponding differential equations. Several remarkable examples are provided.

Notes on a conformal characterization of $2$-dimensional Lorentzian manifolds with constant Ricci scalar curvature