<p>In this study, we investigate the coupled nonlinear integrable system known as the Akbota–Gudekli–Kairat–Zhaidary (AGKZ) equation. By employing a generalized extended simple equation method, we examine soliton and various solitary wave solutions with diverse physical structures. The nonlinear complex AGKZ equation is a newly introduced integrable model arising in the study of space curves and surfaces; therefore, its analytical exploration is essential for understanding its physical applications. The investigated solutions display distinct physical structures, including bright solitons, kink wave structures, dark solitons, peakon-type bright and dark waves, anti-kink wave structures, periodic waves with varying profiles, solitary waves through contour plots, two-dimensional plots, and three-dimensional visualizations using Mathematica tool. The novelty of this work lies in establishing enriched and distinct soliton solutions to the AGKZ equation and performing a comparative analysis of the proposed method, which has not been previously addressed in the literature. The derived solutions of the AGKZ equation may be applied to model ultrashort pulse propagation in nonlinear optical fibers, photonic crystals, waveguides, and solitary waves in shallow water. The results demonstrate that the proposed approach is practical, straightforward, and effective for generating a wide variety of soliton solutions applicable to other nonlinear equations.</p>