  We show that if $(\mathbb{T}^{2},g)$ is a class A Lorentzian 2-torus with
timelike poles, then there exists a Lipschitz foliation by complete
future-directed timelike geodesics with any pre-assigned asymptotic direction
in the interior of the stable time cone. This is done by constructing certain
$C^{1,1}$ solutions to the equation $g(\nabla u,\nabla u)=-1$ on the Abelian
cover $(\mathbb{R}^{2},g)$.
