On class A Lorentzian 2-tori with poles I: Closed geodesics pass through poles

In this paper, by studying certain isometries on globally hyperbolic planes, we prove that if $p$ is a timelike pole on a class A Lorentzian 2-torus, then there exists a closed timelike geodesic passing through $p$ with any preassigned free homotopy class in the interior of the stable time cone. We also show a non-rigid result when timelike poles appear.