  We consider one-dimensional, locally finite interacting particle systems with
two conservation laws which under Eulerian hydrodynamic limit lead to
two-by-two systems of conservation laws:
  \pt \rho +\px \Psi(\rho, u)=0
  \pt u+\px \Phi(\rho,u)=0,
  with $(\rho,u)\in{\cal D}\subset\R^2$, where ${\cal D}$ is a convex compact
polygon in $\R^2$. The system is typically strictly hyperbolic in the interior
of ${\cal D}$ with possible non-hyperbolic degeneracies on the boundary
$\partial {\cal D}$. We consider the case of isolated singular (i.e. non
hyperbolic) point on the interior of one of the edges of ${\cal D}$, call it
$(\rho_0,u_0)=(0,0)$ and assume ${\cal D}\subset\{\rho\ge0\}$. This can be
achieved by a linear transformation of the conserved quantities. We investigate
the propagation of small nonequilibrium perturbations of the steady state of
the microscopic interacting particle system, corresponding to the densities
$(\rho_0,u_0)$ of the conserved quantities. We prove that for a very rich class
of systems, under proper hydrodynamic limit the propagation of these small
perturbations are \emph{universally} driven by the two-by-two system
  \pt\rho + \px\big(\rho u\big)=0
  \pt u + \px\big(\rho + \gamma u^2\big) =0
  where the parameter $\gamma:=\frac12 \Phi_{uu}(\rho_0,u_0)$ (with a proper
choice of space and time scale) is the only trace of the microscopic structure.
The proof is valid for the cases with $\gamma>1$.
  [truncated]
