Some new types of exact solutions for the Kac–Wakimoto equation associated with $e_6^{\left(1\right)}$

Recently, it is shown that the Kac–Wakimoto equation associated with $e_6^{\left(1\right)}$ is not integrable since it does not pass Painlevé test and does not have three-soliton solution, even it has one- and two-soliton solutions. Thus in this paper, we investigate some new types of exact solutions for this equation based on its bilinear form. As a result, the rational solutions, kink-type breather solution and degenerate three-solitary wave solutions of this equation are found. The properties and space structures of these exact solutions are analyzed by displaying their profiles in (x, y)-directions. Furthermore, the Lie symmetry analysis is done to present the one-parameter group of symmetries for the KW equation. (paper)