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By employing the generalized bilinear method and symbolic computation, we obtain new exact solutions of a generalized Calogero–Bogoyavlenskii–Schiff​ equation, which include breather wave solutions, periodic lump soliton solutions, and rational solutions. Via the three-dimensional plot and density plot with the help of Maple, the physical characteristics of these waves are described very well. These outcomes will serve as a very significant milestone in the study of water waves in mathematical physics and engineering phenomena.