Classification of compact Lorentzian $2$-orbifolds with noncompact full isometry groups

Among closed Lorentzian surfaces, only flat tori can admit noncompact full isometry groups. Moreover, for every $n\ge3$ the standard $n$-dimensional flat torus equipped with canonical metric has a noncompact full isometry Lie group. We show that this fails for $n=2$ and classify the flat Lorentzian metrics on the torus with a noncompact full isometry Lie group. We also prove that every two-dimensional Lorentzian orbifold is very good. This implies the existence of a unique smooth compact $2$-orbifold, called the pillow, admitting Lorentzian metrics with a noncompact full isometry group. We classify the metrics of this type and make some examples.