  Applying the general theory about complete spacelike stationary (i.e. zero
mean curvature) surfaces in 4-dimensional Lorentz space $\mathbb{R}^4_1$, we
classify those regular algebraic ones with total Gaussian curvature $-\int
K\mathrm{d}M=4\pi$. Such surfaces must be oriented and be congruent to either
the generalized catenoids or the generalized enneper surfaces. For
non-orientable stationary surfaces, we consider the Weierstrass representation
on the oriented double covering $\widetilde{M}$ (of genus $g$) and generalize
Meeks and Oliveira's M\"obius bands. The total Gaussian curvature are shown to
be at least $2\pi(g+3)$ when $\widetilde{M}\to\mathbb{R}^4_1$ is
algebraic-type. We conjecture that there do not exist non-algebraic examples
with $-\int K\mathrm{d}M=4\pi$.
