Branching rules for irreducible supercuspidal representations of unramified $\mathrm{U}(1,1)$

Let $G$ denote the unramified quasi-split unitary group $\mathbb{U}(1,1)(F)$ over a $p$-adic field $F$ with residual characteristic $p\mathrm{\ne }2$. In this article, we determine the branching rules for all irreducible supercuspidal representations of $G$, that is, we explicitly describe their decomposition upon restriction to a fixed maximal compact subgroup $\mathcal{K}$ in terms of irreducible representations of $\mathcal{K}$. We also present two applications of these decompositions, which verify two new conjectures in the literature for $G$. Together with the results from a previous paper by the author, this article completes the description of the branching rules for all irreducible smooth representations of $G$.