The five-component Manakov system (alias 5-Manakov system) appears in many fields such as nonlinear optics, Bose–Einstein condensates (alias the spin-2 Gross–Pitaevskii equation), and ocean. In this Letter, we investigate the semi-rational vector rogon–soliton and soliton of the 5-Manakov system with mixed zero and non-zero backgrounds by using the modified Darboux transform. The key point is to explicitly solve the multiple roots of a sixth degree characteristic polynomial equation. In particular, we exhibit the wave structures of the fundamental semi-rational vector rogon with grey solitons and soliton solutions, and their higher-degree forms. Moreover, the semi-rational vector rogon with grey solitons corresponding to the components with non-zero backgrounds are $\mathcal{P}T$-symmetric. These semi-rational wave structures will be useful to further understand some related physical phenomena and to design the relative experiments.

五分量马纳科夫系统（又名5-马纳科夫系统）出现在非线性光学、玻色-爱因斯坦凝聚（又名spin-2 Gross-Pitaevskii方程）、海洋等诸多领域。在这封信中，我们使用改进的 Darboux 变换研究了具有混合零和非零背景的 5-Manakov 系统的半有理向量 rogon-soliton 和孤子。关键是明确求解六阶特征多项式方程的多重根。特别是，我们展示了具有灰色孤子和孤子解的基本半有理向量 rogon 的波结构，以及它们的高次形式。此外，与非零背景分量对应的具有灰色孤子的半有理向量 rogon 是$\mathcal{磷}吨$-对称。这些半有理波结构将有助于进一步了解一些相关的物理现象和设计相关的实验。