Hyperelliptic tangential covers and even elliptic finite-gap potentials, back and forth

Let $\pi\colon (\Gamma,p) \to (X,\omega_0)$ denote a degree-$n$ cover of an elliptic curve, marked at a smooth $p\in \Gamma$. Consider the (rational) Abel map $\mathrm{Ab}_p\colon \Gamma \to \operatorname{Jac}\Gamma$ and the dual map $\pi^\vee:= \mathrm{Ab}_p\circ\pi^* \colon X\to\operatorname{Jac}\Gamma$ into the Jacobian of $\Gamma$. We call $\pi$ a hyperelliptic tangential cover (HT-cover) if $\Gamma$ is a hyperelliptic curve, $p\in \Gamma$ is a Weierstrass point and the images of $\Gamma$ and $X$ in $\operatorname{Jac}\Gamma$ are tangent at the origin. To any such HT-cover $\pi$ we attach an integer vector $\mu \in \mathbb{N}^4$, the so-called type, satisfying $\mu_0+1\equiv\mu_1\equiv\mu_2\equiv\mu_3\equiv n \ \operatorname{mod}2$ and $2n+1-\sum_i \mu_i^2=4d$ for some $d\in \mathbb{N}$. Whenever $\Gamma$ is smooth, the type $\mu$ gives the number of Weierstrass points of $\Gamma$ (different from $p$) over each half-period $\omega_i$ of $X$, $i=0,\dots,3$. We denote by $\mathcal{S}\mathcal{C}_X(\mu,d)$ the set of degree-$n$ HT-covers of type $\mu$. Then the even, doubly-periodic finite-gap potential associated with $\mathcal{S}\mathcal{D}_X(\mu,d)=\{(\pi,\xi)\colon\pi\in\mathcal{S}\mathcal{C}_X(\mu,d),\,\xi\text{ is a theta characteristic of }\pi\}$, decompose as
\( u_\xi(x)=\sum_0^3\alpha_i(\alpha_i+1)\wp(x-\omega_i) +2\sum_{j=1}^m \bigl(\wp(x-\rho_j)+\wp(x+\rho_j)\bigr) \)
for some $(\alpha,m)\in \mathbb{N}^4\times \mathbb{N}$ such that $2n=\sum_i\alpha_i(\alpha_i+1)+4m$.
The set $\mathcal{P}ot_X(\alpha,m)$ of such potentials is finite, and we have a bijection
\( (\pi,\xi)\in\bigcup_{(\mu,d)}\mathcal{S}\mathcal{D}_X(\mu,d)\mapsto u_\xi\in\bigcup_{(\alpha,m)}\mathcal{P}ot_X(\alpha,m). \)

The problem at stake is to find the inverse map, as well as the cardinals $\#\mathcal{S}\mathcal{C}_X(\mu,d)$ and $\#\mathcal{P}ot_X(\alpha,m)$. The latter problem has been thoroughly studied for $\mathcal{P}ot_X(0)$ and $\mathcal{P}ot_X(1)$. We prove that $\#\mathcal{P}ot_X(\alpha,2)= 27$ for a generic elliptic curve $X$ and find the inverse image of $\mathcal{P}ot_X(\alpha,m)$. Bounds for the types of arithmetic genera of the spectral data for elements follow. We conclude with a conjectural recursive formula on $d\in \mathbb{N}$ for $\#\mathcal{P}ot_X(\alpha,d)$ and $\#\mathcal{S}\mathcal{C}_X(\mu,d)$.