  Kahan discretization is applicable to any quadratic vector field and produces
a birational map which approximates the shift along the phase flow. For a
planar quadratic Hamiltonian vector field with a linear Poisson tensor and with
a quadratic Hamilton function, this map is known to be integrable and to
preserve a pencil of conics. In the paper `Three classes of quadratic vector
fields for which the Kahan discretization is the root of a generalised Manin
transformation' by P. van der Kamp et al., it was shown that the Kahan
discretization can be represented as a composition of two involutions on the
pencil of conics. In the present note, which can be considered as a comment to
that paper, we show that this result can be reversed. For a linear form
$\ell(x,y)$, let $B_1,B_2$ be any two distinct points on the line
$\ell(x,y)=-c$, and let $B_3,B_4$ be any two distinct points on the line
$\ell(x,y)=c$. Set $B_0=\tfrac{1}{2}(B_1+B_3)$ and $B_5=\tfrac{1}{2}(B_2+B_4)$;
these points lie on the line $\ell(x,y)=0$. Finally, let $B_\infty$ be the
point at infinity on this line. Let $\mathfrak E$ be the pencil of conics with
the base points $B_1,B_2,B_3,B_4$. Then the composition of the
$B_\infty$-switch and of the $B_0$-switch on the pencil $\mathfrak E$ is the
Kahan discretization of a Hamiltonian vector field
$f=\ell(x,y)\begin{pmatrix}\partial H/\partial y \\ -\partial H/\partial x
\end{pmatrix}$ with a quadratic Hamilton function $H(x,y)$. This birational map
$\Phi_f:\mathbb C P^2\dashrightarrow\mathbb C P^2$ has three singular points
$B_0,B_2,B_4$, while the inverse map $\Phi_f^{-1}$ has three singular points
$B_1,B_3,B_5$.
