Hyperelliptic $d$-osculating covers and rational surfaces

Let ${\mathbb{P}}^1$ and $(X,q)$ denote, respectively, the projective line and a fixed elliptic curve marked at its origin, both defined over an algebraically closed field $\mathbb{K}$ of arbitrary characteristic $\emph{$\mathbf{\text{p}}$} \neq2$\mathrm{.}Wewillconsiderallfiniteseparablemarkedmorphisms$\pi :(\Gamma ,p)\rightarrow (X,q)$,suchthat$\Gamma$isadegree-$2$coverof$ $\mathbb{P}$^1$,ramifiedatthesmoothpoint$p \in \Gamma$. Canonically associated to $\pi$ there is the Abel (rational) embedding of $\mathrm{\Gamma }$ into its \emph{generalized Jacobian}, $A_p:\mathrm{\Gamma }\to Jac\mathrm{\Gamma }$, and ${0} \subsetneq V^1_{\Gamma,p}…\subsetneq V ^g_{\Gamma,p}$, the flag of hyperosculating planes to $A_p(\mathrm{\Gamma })$ at $A_p(p)\in Jac\mathrm{\Gamma }$ (cf. \textbf{2.1. & 2.2.}). On the other hand, we also have the homomorphism ${\iota }_{\pi }:X\to \text{Jac}\mathrm{\Gamma }$, obtained by dualizing $\pi$. There is a smallest positive integer $d$ such that the tangent line to ${\iota }_{\pi }(X)$ is contained in $V_{\mathrm{\Gamma },p}^d$. We call it \emph{the osculating order} of $\pi$. Studying, characterizing and constructing those with given \textit{osculating order} $d$ but maximal possible arithmetic genus, is one of the main issues. The other one, to which the first issue reduces, is the construction of all rational curves in a particular anticanonical rational surface associated to $X$ (i.e.: a rational surface with an effective anticanonical divisor).