This paper presents a comprehensive investigation into the complex behavior of the Kuralay-II equation. We begin by analyzing the bifurcation structure and corresponding phase portraits of the equation to gain insight into the underlying dynamical transitions. Next, we examine the chaotic behavior of the governing system through a Lyapunov stability analysis, which provides valuable insights into the sensitivity and stability characteristics of the model. Additionally, we derive several explicit solutions to the equation using an exact analytical method and visualize some of these solutions to highlight their structural properties and dynamical implications. The findings provide a solid framework for future research aimed at understanding the complex behavior of nonlinear systems.