Let $X$ be a compact Riemann surface of genus $g\ge 2$. The moduli space $\mathcal{M}(r,d)$ of rank $r$ and degree $d$ semi-stable Higgs bundles over $X$ admitted a stratification, called Shatz stratification, which was defined by the Harder-Narasimhan type of the Higgs bundles. There was also a ${\mathbb{C}}^{\ast }$-action on $\mathcal{M}(r,d)$ given by the product on the Higgs field, which provided the Białynicki-Birula stratification by considering the Hodge limit bundles $\underset{z\to 0}{lim}(E,z\cdot \phi )$. In this paper, these limit bundles were computed for all possible Harder-Narasimhan types when the rank of the Higgs bundles was $r=5$, explicit vector forms were provided for the Hodge limit bundles, and necessary and sufficient conditions were given for them to be stable. In addition, it was proved that, in rank $5$, the Shatz strata traversed the Białynicki-Birula strata. Specifically, it was checked that there existed different semi-stable rank $5$ Higgs bundles with the same Harder-Narasimhan type such that their associated Hodge limit bundles were not S-equivalent, and explicit constructions of those Higgs bundles were also provided.

Let $X$ be a compact Riemann surface of genus $g\ge 2$ and $M(G_2)$ be the moduli space of polystable principal $G_2$-bundles over $X$. The Harder-Narasimhan types of the bundles induced a stratification of the moduli space $M(G_2)$ called Shatz stratification. In this paper, a description of the Shatz strata of the unstable locus of $M(G_2)$ corresponding to certain family of Harder-Narasimhan types (specifically, those of the form $(\lambda ,\mu ,0,-\mu ,-\lambda )$ with $\mu <\lambda \le 0$) was given. For this purpose, a family of vector bundles was constructed in which a 3-form and a 2-form were defined so that it was proved that they were strictly polystable principal $G_2$-bundles. From this, it was proved that, when the genus of $X$ was $g\ge 12$, these Shatz strata were the disjoint union of a family of $G_2$-Hecke curves in $M(G_2)$ that will be constructed along the paper. Therefore, the presented results provided an advance in the knowledge of the geometry of $M(G_2)$ through the study of its Shatz strata and presented a methodological innovation, by using Hecke curves for this study.