Let $X$ be a del Pezzo surface of degree one over a field $k$ of characteristic not $2$ equipped with a conic bundle $\pi: X \rightarrow \mathbb{P}^1$. By Theorem 5.6 of this paper, $X$ admits a closed embedding into the space $\mathbb{F}(0,1,1)$ as a surface of bidegree $(1,2)$. That is, we may write
$$ X: A(s,t)x_0^2 + B(s,t)x_0x_1 + C(s,t)x_0x_2 + D(s,t)x_1^2 + E(s,t)x_1x_2 + F(s,t)x_2^2 = 0 \subset \mathbb{F}(0,1,1) $$ with $$ \deg A = 1, \quad \deg B = \deg C = 2, \quad \deg D = \deg E = \deg F = 3, $$
where one identifies $\mathbb{F}(0,1,1)$ with the quotient $((\mathbb{A}^2_{s,t} \setminus \{0\}) \times (\mathbb{A}^3_{x_0,x_1,x_2} \setminus \{0\}))/\mathbb{G}_m^2$ with the action of $(\lambda,\mu) \in \mathbb{G}_m^2$ given by $(\lambda,\mu) * ((s,t),(x_0,x_1,x_2)) = ((\lambda s,\lambda t),(\mu x_0, \lambda^{-1} \mu x_1,\lambda^{-1} \mu x_2))$. The necessary background on projective bundles and conic bundles can be found in Section 2 of the linked paper.
On the other hand, by considering the values of $h^0(-nK_X)$ for $1 \leq n \leq 6$, a del Pezzo surface of degree one can be embedded as a sextic hypersurface in the weighted projective space $\mathbb{P}(1,1,2,3)$. In addition to assuming $\text{char} k \neq 2$, let me further assume $\text{char} k \neq 3$. Then we may embed $X$ as
$$ X: w^2 = z^3 + g(x,y)z + h(x,y) \subset \mathbb{P}(1,1,2,3) $$ for $g,h \in k[x,y]$ of degrees $4$ and $6$ respectively.
In this form, we see that $X$ is a double cover of a quadric cone branched along a sextic curve. The Bertini involution $\iota: X \rightarrow X$ is the involution swapping the two sheets of this cover, i.e. $\iota((x,y,z,w)) = (x,y,z,-w)$.
My main question is as follows.
Question 1: Given an equation for $X$ of the first form, how can we find an equation for $X$ of the second form?
However, I would be very happy to have an answer to an ostensibly easier question.
Question 2: How can one express the Bertini involution when $X$ is in the first form?
Regarding Question 1: I was hoping to do this by considering the anticanonical map $X \dashrightarrow \mathbb{P}^1$. In the first form, it is projection onto $\mathbb{P}^1_{x_1,x_2}$. In the latter form, it is projection onto $\mathbb{P}^1_{x,y}$. Indeed, in both cases, the given coordinates are generators of $H^0(-K_X)$. The second form essentially expresses the curves in $|-K_X|$, which are of genus one, in Weierstrass form, so I was looking to find a way to rewrite the fibres of $X \dashrightarrow \mathbb{P}^1_{x_1,x_2}$ in Weierstrass form. The problem is that I do not know the coordinates for my Weierstrass form: $z \in H^0(-2K_X)$ and $w \in H^0(-3K_X)$. You could view Question 1 as asking how to find the "extra" global section $z$ in $H^0(-2K_X)$ (in addition to $x_1^2$, $x_1x_2$ and $x_2^2$) and then $w$ in $H^0(-3K_X)$ (which is not a monomial of weighted degree $3$ in $x_1$, $x_2$ and $z$).
Regarding Question 2: The Bertini involution coincides, along smooth anticanonical curves $E$, with multiplication by -1 on the elliptic curve $E$ with origin the unique base-point of $|-K_X|$. This can be seen from the second form. Can one use this to write down what the Bertini involution does in the first form?
