<p>In the devoted study, the Lie symmetry method has been applied to the nonlinear Calogero–Degasperis equation to construct exact closed-form solutions, which study examines the interaction between Riemann waves propagating along the y-axis and long waves propagating along the x-axis. The Lie group method is recruited to deduce infinitesimal generators under invariance conditions of one-parameter transformations. The independent variables turned out to be diminishable under infinitesimal invariance conditions, meanwhile conserving the characteristics of the system. The infinite-dimensional algebra of symmetries has been exhibited by the commutative relations of infinitesimal vectors. Subsequently, similarity variables have been derived by employing infinitesimal generators, which steer to first symmetry reductions. Thereupon, an equivalent system of ordinary differential equations has been furnished by a repeated process of symmetry reductions. In the next stage, we constructed conservation vectors with associated symmetries under Lagrangian formulations. Moreover, the dynamics of every solution have been discussed via their evolution profiles. By taking arbitrary functions and constants involved in the solutions, the numerical simulations have been executed. The results of this study are significant for understanding how solitons move around in physical systems. The derived solutions are more general than previously accepted results and have never been reported.</p>