On class A Lorentzian 2-tori with poles II: Foliations by timelike lines

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 $(\mathbb{R}^{2},g)$.