The \((2+1)\) -dimensional Broer–Kaup equations model the movement of long, dispersive gravity waves travelling in opposite directions within a body of water of constant depth. This system has significant implications across various scientific fields, such as plasma physics and nonlinear optical fibre communications. In this paper, we employed a classical Lie symmetry analysis to investigate the analytical solutions and soliton behaviour of the equations. To highlight the originality of our work, we compared our results with previous studies. The authors emphasise that no one could have obtained such a new class of solutions as those derived in this study without restricting all arbitrary functions involved in infinitesimal test problems. The authors did not apply any restrictions to \(f_1 (y)\) and \(f_2 (t)\) , and \(f_3 (t)\) is chosen as \(\frac{a_0}{2}f'_2(t)\) (where \(a_0 \ne 0\) is a constant for further integration), which increases the generality of the answers and provides additional opportunities to describe physical occurrences. To further demonstrate the integrability of the (2+1)-coupled Broer–Kaup equations (CBKEs) (1), conserved vectors were also utilised. We used the Lie symmetry method to change the original set of partial differential equations into a similar set of ordinary differential equations that are limited in a certain way. This procedure made integration easier. Our examination of soliton dynamics provides valuable insights into the physical characteristics of the solutions. Additionally, we utilised conserved vectors to demonstrate the integrability of the system. The outcomes of this research significantly enhance the practical applications of the Broer–Kaup equations.
