Timelike surfaces with zero mean curvature in Minkowski 4-space

On any timelike surface with zero mean curvature in the four-dimensional Minkowski space we introduce special geometric (canonical) parameters and prove that the Gauss curvature and the normal curvature of the surface satisfy a system of two natural partial differential equations. Conversely, any two solutions to this system determine a unique (up to a motion) timelike surface with zero mean curvature so that the given parameters are canonical. We find all timelike surfaces with zero mean curvature in the class of rotational surfaces of Moore type. These examples give rise to a one-parameter family of solutions to the system of natural partial differential equations describing timelike surfaces with zero mean curvature.