Classification of zero mean curvature surfaces of separable type in Lorentz-Minkowski space

Consider the Lorentz-Minkowski $3$-space $\mathbb{L}^3$ with the metric $dx^2+dy^2-dz^2$ in canonical coordinates $(x,y,z)$. A surface in $\mathbb{L}^3$ is said to be separable if satisfies an equation of the form $f(x)+g(y)+h(z)=0$ for some smooth functions $f$, $g$ and $h$ defined in open intervals of the real line. In this article we classify all zero mean curvature surfaces of separable type, providing a method of construction of examples.