  On the Abelian cover $(\mathbb{R}^{2},g)$ of a class A Lorentzian 2-torus
$(\mathbb{T}^{2},g)$, we showed the existence of global viscosity solutions to
the eikonal equation
  $$ g(\nabla u,\nabla u)=-1
  $$ associated to those homologies in the interior of the homology cone. Some
other related dynamical properties are also considered. As an application of
the main results, we study the differentiability of the unit sphere of the
stable time separation associated to the class A Lorentzian 2-torus.
