We consider the H-BMSS-γ system published in [Escalante et al., 2019]. It is a shallow water-like system which captures non-hydrostatic pressure effects. In addition to that, it is a non-conservative hyperbolic system. We consider two types of solutions to the H-BMSS-γ system: firstly, we look at a solitary wave, for which we suggest better values for the constants compared to the already existing solutions. Secondly, we compute the solution to Riemann Problem without bathymetry, but also in the dry case. Furthermore, we implement a numerical discretization of the H-BMSS-γ system using both a finite volume method, and an ADER-DG (Adaptive DERivative Discontinuous
Galerkin) method. The ADER-DG method which we implemented uses an additional a posteriori limiter with our finite volume method, in order to handle discontinuities better. We also construct four well-balanced fluxes for said system, and compute a Roe average matrix. Finally, we show experimentally that our scheme observes the Resting Lake Property, the convergence of it to reference solutions. Additionally, we verify that it conforms to our two solutions to the H-BMSS-γ system.