<p>The coupled mKdV–Calogero–Bogoyavlenskii–Schiff equations are utilized to examine nonlinear wave interactions in fluids, particularly demonstrating nonlinearity at equilibrium attributed to cubic nonlinearity. This research article offers analytical solutions derived from optimal subalgebra classification and classical Lie symmetry analysis. The profiles of positons and negatons were examined utilizing the prominently visible animation framework. Comparisons of the acquired results with prior findings demonstrate the originality of the outcomes. We employ Noether’s theorem to determine the conserved vectors of the system. The solutions may demonstrate their applicability in identifying invariant solutions to nonlinear partial differential equations and in comparing numerical outcomes.</p>