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We show that if $(\mathbb{T}^{2},g)$ is a class A Lorentzian 2-torus withtimelike poles, then there exists a Lipschitz foliation by completefuture-directed timelike geodesics with any pre-assigned asymptotic directionin 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 Abeliancover $(\mathbb{R}^{2},g)$.

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