Marginally trapped surfaces in ${\mathbb{L}}^{4}$ and three Weierstrass representations

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.