We study $2$-dimensional unit vector flows on graphs, that is, nowhere-zero flows that assign to each oriented edge a unit vector in $\mathbb R^{3}$. We give a new geometric characterization of $\mathbb S^{2}$-flows on cubic graphs. We also prove that the class of cubic graphs admitting an $\mathbb S^{2}$-flow is closed under a natural composition operation, which yields further constructions; in particular, blowing up a vertex into a triangle preserves the existence of an $\mathbb S^{2}$-flow. Our second contribution is algebraic: we extend the rank-based approach of [SIAM J. Discrete Math., 29 (2015), pp.~2166--2178] from $\mathbb S^{1}$-flows to $\mathbb S^{2}$-flows. More precisely, we show that if an $\mathbb S^{2}$-flow $\varphi$ satisfies $\operatorname{rank}(S_{\mathbb{Q}}(\varphi))\le 2$ and $S_{\mathbb{Q}}(\varphi)$ is odd-coordinate-free, then the graph admits a nowhere-zero $4$-flow.

翻译：我们研究图上的二维单位向量流，即无处为零的流，它将每个有向边赋值为$\mathbb R^{3}$中的一个单位向量。我们给出了三次图上$\mathbb S^{2}$流的一种新的几何刻画。我们还证明了允许$\mathbb S^{2}$流的三次图类在一种自然的合成操作下是封闭的，这产生了进一步的构造；特别地，将一个顶点膨胀为一个三角形会保持$\mathbb S^{2}$流的存在性。我们的第二个贡献是代数方面的：我们将[SIAM J. Discrete Math., 29 (2015), pp.~2166--2178]中基于秩的方法从$\mathbb S^{1}$流推广到$\mathbb S^{2}$流。更准确地说，我们证明如果一个$\mathbb S^{2}$流$\varphi$满足$\operatorname{rank}(S_{\mathbb{Q}}(\varphi))\le 2$且$S_{\mathbb{Q}}(\varphi)$是奇坐标自由的，则该图允许一个无处为零的$4$-流。