Parametrizations of minimal timelike surfaces in the four-dimensional pseudo-Euclidean space of index two

We construct representation formulas for local null curves in the four-dimensional pseudo-Euclidean space of index two and derive corresponding parametrizations for local minimal timelike surfaces without integration. As a special case of the representation formula, we construct a representation formula for local null curves in the three-dimensional pseudo-Euclidean space of index one that involves integration. Our results provide examples of minimal timelike surfaces.