I am looking for explicit equations for rational elliptic surfaces in characteristic $2$. For me, a rational elliptic surface $X$ is a smooth projective surface $X$ which is rational and equipped with a morphism $X\to \mathbb{P}^1$ whose general fibres are elliptic curves. I work over an algebraically closed field $k$ of characteristic $2$.
In the article "Configurations of singular fibres on rational elliptic surfaces in characteristic two", William E. Lang says that he assumes the surface to be relatively minimal (so exceptional curves in the fibres) and then obtains an equation of the form
$$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$$
where $a_i\in k[t]$ is a polynomial of degree $\le i$ in $t$. This is then an affine version of the surface, that we can see in $\mathbb{A}^3$ with coordinates $x,y,t$.
Do we always have this kind of equations? Why? And are all such equations giving rational elliptic surfaces?
Note that the pencil of elliptic curves can be put up to birational maps into a pencil of curves of degree $3m$ with multiplicity $m$ at $9$ points (Halphen pencil). As the integer $m$ is uniquely determined by the pencil, it seems strange to me that we can bound the degree of the equations of the $a_i$ when $m$ is large enough. Or is the equation only OK for $m=1$ ? If yes, then what would be equations for $m=2$, $m=3$ or $m=4$ ?
