  Minimal surfaces with planar curvature lines in the Euclidean space have been
studied since the late 19th century. On the other hand, the classification of
maximal surfaces with planar curvature lines in the Lorentz-Minkowski space has
only recently been given. In this paper, we use an alternative method not only
to refine the classification of maximal surfaces with planar curvature lines,
but also to show that there exists a deformation consisting exactly of all such
surfaces. Furthermore, we investigate the types of singularities that occur on
maximal surfaces with planar curvature lines. Finally, by considering the
conjugate of maximal surfaces with planar curvature lines, we obtain analogous
results for maximal surfaces that are also affine minimal surfaces.
