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

In this paper, we show that if (T2,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 C1,1 solutions to the equation g(∇u,∇u)=−1 on the Abelian cover (R2,g).