  A connected regular surface in Lorentz-Minkowski 3-space is called a mixed
type surface if the spacelike, timelike and lightlike point sets are all
non-empty. Lightlike points on mixed type surfaces may be regarded as singular
points of the induced metrics. In this paper, we introduce the L-Gauss map
around non-degenerate lightlike points, and show the fundamental theorem of
surface theory for mixed type surfaces at non-degenerate lightlike points. As
an application, we prove that a real analytic mixed type surface admits
non-trivial isometric deformations around generic lightlike points.
