Abstract
The current study deals with investigation of Einstein's vacuum field equation for exploring movable critical points. We employ first the Painlevé analysis, and then we use the auto-Bäcklund transformation. Moreover, the Lie classical method will be implemented to obtain similarity reductions and exact solutions via discovering the entire sets of point symmetries. We show that symmetries of Einstein's vacuum field equation form an infinite-dimensional Lie algebra and arbitrary function in acquired solutions. In addition, various other arbitrary parameters provide enough freedom to simulate physical situations governed by this equation are observed.
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