In this article, a (2+1)-dimensional system of Broer–Kaup–Kupershmidt (BKK) equations, which describes the non-linear, and long gravity waves in a dispersive system, is investigated by applying the method of Lie group of invariance. Applying the Lie group technique, the infinitesimals, vector fields, commutator table, and adjoint table are constructed for the BKK system. Furthermore, a one-dimensional optimal system of subalgebras is determined with the help of the adjoint transformation matrix; thereafter, the BKK system of equations is reduced into many systems of ordinary differential equations (ODEs) with respect to the similarity variables obtained through the symmetry reduction. These systems of ODEs are solved under some parametric constraints to obtain the exact closed form solutions. The obtained results are interpreted physically via graphical representation. Thus, dark–bright solitary waves, multi-peak mixed waves, breather type waves, periodic waves, multi-peakon solitons, and multi-solitons profiles of the obtained solutions are presented to make this research physically significant. A comparison is also presented between the solutions obtained in this article and the solutions reported by Kassem and Rashed in Kassem and Rashed (2019). The solutions obtained in this article are more general, and completely different in view of solutions obtained by Kassem and Rashed. The resulting solutions are found to be useful to understand the dynamics of BKK model and represent the authenticity as well as the effectiveness of the Lie group method. Therefore, the obtained solutions, and their dynamical wave structures are quite significant for understanding the propagation of the long gravity waves in a dispersive system. To obtain the infinitesimal generators and wave profiles of obtained solutions, symbolic computation is performed in the software package MAPLE and MATHEMATICA.

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