The Kuralay-Ma-Myrzakulov equation (KMME) and the Akbota- Myrzakulov-Tolkynay-Zhaidary equation (AMTZE) are studied. The integrability of these two equations is achieved by the existence of corresponding Lax pairs. The corresponding Lax representations are presented. Gauge equivalence between these two integrable equations is proved. It is shown that the KMME admits two integrable reductions, namely, the Manukure-Zhanbota equation and the Manukure-Zhaidary equation. Similarly, the AMTZE has two integrable reductions, namely, the Kairat-Kuralay-Myrzakulov-Shynaray equation (KKMSE) and the Wu-Zhang equation (WZE). From these results, it follows that the Manukure- Zhanbota equation is gauge equivalent to the KKMSE. At the same time, the Manukure-Zhaidary equation and the Wu-Zhang equation are gauge equivalent to each other. Some exact traveling wave solutions of the AMTZE are presented. These solutions demonstrate a variety of structures, including Jacobi elliptic, trigonometric, soliton, and rational forms. The results are illustrated through 3D and contour plots, which clearly depict the system's behavior during momentum propagation and help identify suitable parameter values. This graphical analysis offers important insights into the properties and dynamics of the soliton solutions derived from the integrable AMTZE equation.

In this paper, we study an integrable system with the self-consistent potentials called the Nurshuak-Tolkynay-Myrzakulov (NTM) system. This system is of great importance in the theory of integrable nonlinear equations, since this system describes the dynamics of nonlinear wave processes in various fields of physics, such as hydrodynamics, optics, quantum mechanics, and plasma dynamics. Various integrable reductions of this system are also given and their Lax pairs are found. It is shown that the NTM system, being integrable, has some deep geometric roots, and that its geometric interpretation can lead to an understanding of more complex geometric structures. Thus, it is shown that the NTM system describes the dynamics of waves and allows us to understand how those waves interact with the geometry of space, which is an important aspect of many physical processes. Solitonic solutions of the NTM system are found. These solutions exhibit various signs of the periodicity, exponentiality, and rationality of soliton structures, including the elliptic Jacobi function. The results are visualized using three-dimentional (3D) and contour plots to clearly illustrate the response of the behavior to momentum propagation and to find appropriate values for the system's parameters. This visualization provides valuable insights into the characteristics and dynamics of the soliton solutions obtained from the integrable NTM equation.