  We construct new integrable systems to present Weierstrass type
representations for spacelike surfaces whose mean curvature vector $\mathbf{H}$
satisfies the null condition $\langle \mathbf{H}, \mathbf{H} \rangle=0$ in the
four dimensional Lorentz-Minkowski space ${\mathbb{L}}^{4}$. Our new
Weierstrass presentations extend simultaneously classical Weierstrass
representations (of the first kind and the second kind) for maximal surfaces in
${\mathbb{L}}^{3}$ and minimal surfaces in ${\mathbb{R}}^{3}$. We solve a
linear partial differental equation to construct explicit examples of
marginally trapped surfaces with nowhere vanishing mean curvature vector.
