Branching rules for the restriction of regular representations of ${\mathrm{G}\mathrm{L}}_2(\mathfrak{o}\mathrm{/}{\mathfrak{p}}^r)$ to ${\mathrm{S}\mathrm{L}}_2(\mathfrak{o}\mathrm{/}{\mathfrak{p}}^r)\mathrm{.}$

Let $\mathfrak{o}$ be a compact discrete valuation ring with maximal ideal $\mathfrak{p}$ such that the finite residue field $\mathfrak{o}\mathrm{/}\mathfrak{p}$ has characteristic $p\mathrm{.}$ For $r\ge 2$ and $p=2,$ we obtain the branching rules for the restriction of a regular representation of ${\mathrm{G}\mathrm{L}}_2(\mathfrak{o}\mathrm{/}{\mathfrak{p}}^r)$ to ${\mathrm{S}\mathrm{L}}_2(\mathfrak{o}\mathrm{/}{\mathfrak{p}}^r)\mathrm{.}$ These results have different behaviour than that of the known case of $p\mathrm{\ne }2.$