  With several concrete examples of zero mean curvature surfaces in
$\boldsymbol{R}^3_1$ containing a light-like line recently having been found,
here we construct all real analytic germs of zero mean curvature surfaces by
applying the Cauchy-Kovalevski theorem for partial differential equations. A
point where the first fundamental form of a surface degenerates is said to be
light-like. We also show a theorem on a property of light-like points of a
surface in $\boldsymbol{R}^3_1$ whose mean curvature vector is smoothly
extendable. This explains why such surfaces will contain a light-like line when
they do not change causal types. Moreover, several applications of these two
results are given.
