  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.
