Causal characters of zero mean curvature surfaces of Riemann type in the Lorentz-Minkowski 3-space

A zero mean curvature surface in the Lorentz-Minkowski 3-space is said to be of Riemann-type if it is foliated by circles and at most countably many straight lines in parallel planes. We classify all zero mean curvature surfaces of Riemann-type according to their causal characters, and as a corollary, we prove that if a zero mean curvature surface of Riemann-type has exactly two causal characters, then the lightlike part of the surface is a part of a straight line.