The Anti-de Sitter solution is one precise solution of Einstein's equations in empty space . One can consider AdS spaces with several dimensions but what important for us is an AdS space with two spatial dimensions plus one of time; AdS 3 Reviewing some of previous works about geometry of AdS 3 spaces, we generalize a notation, that is very similar to Dirac notation, and present the coordinates of the ambient spacetime, SO(2,2), in a ket.Then we introduce the Killing vectors of AdS 3 in spacetime SO(2,2) and peresent one of several radius , under the condition of Anti-de Sitter". Next, we obtain a general metric for the AdS 3 spaces with two Killing vector; the matrix of this metric have six non-zero component.Then we dissect conditions for diagonalizing the metric and show that "this conditions are not permanent" and there are some examples with non-zero and components. Last, we introduce a new example with non-zero components, and , obtain it's metric and list the singularity points. Therefore, writing the Lagrangy for the example's metric, we compute the constants of motion and obtain the lightlike geodesies. , in this geometry, is a singularity and we checking if the lightlike geodesies visit it, in slights of proper observer and infinity observer. Additionally , we debate about switch ing the nature of coordinate r when the lightlike geodesies meet some other singularity points. Keywords: Self-dual geometry, Anti-self-dual geometry, three dimensional Anti-de Sitter
