The equivariant genera of marked strongly invertible knots associated with $2$-bridge knots

A marked strongly invertible knot is a triple $(K,h,\delta )$ of a knot $K$ in $S^3$, a strong inversion $h$ of $K$, and a subarc $\delta \subset \mathrm{Fix}(h)\cong S^1$ bounded by $\mathrm{Fix}(h)\cap K\cong S^0$. An invariant Seifert surface for $(K,h,\delta )$ is an $h$-invariant Seifert surface for $K$ that intersects $\mathrm{Fix}(h)$ in the arc $\delta$. In this paper, we completely determine the equivariant genus (the minimum of the genera of invariant Seifert surfaces for $(K,h,\delta )$) of every marked strongly invertible knot $(K,h,\delta )$ with $K$ a $2$-bridge knot.