This research investigates a one-dimensional system of quasi-linear hyperbolic partial differential equations, obtained by vertically averaging the Euler equations between artificial interfaces. This system represents a shallow water model with two velocities and is explored using Lie symmetry analysis to derive several closed-form solutions. Through symmetry analysis, a Lie group of transformations and their corresponding generators are identified via parameter analysis. From these, an optimal one-dimensional system of subalgebras is constructed and classified based on symmetry generators and invariant functions. The model is further simplified by reducing it to ordinary differential equations (ODEs) using similarity variables for each subalgebra, yielding invariant solutions. Additionally, various conservation laws are formulated utilizing the nonlinear self-adjoint property of the governing system. The study concludes by analyzing the behavior of characteristic shocks, $\operatorname{C^1}$-waves, and their interactions, offering a detailed understanding of their dynamics.