The authors extend their earlier compactness framework to some canonical classes of quadratic flux systems with an isolated ombilic point. Using these topological arguments, they establish: (1) the compactness of the solution operator, (2) the long time behaviour in \(L^{\infty}\) of entropy solutions corresponding to large initial data, (3) the convergence of the vanishing viscosity method, of the Lax-Friedrichs scheme and of the Godunov scheme.