  Kahan discretization is applicable to any system of ordinary differential
equations on $\mathbb R^n$ with a quadratic vector field,
$\dot{x}=f(x)=Q(x)+Bx+c$, and produces a birational map $x\mapsto
\widetilde{x}$ according to the formula
$(\widetilde{x}-x)/\epsilon=Q(x,\widetilde{x})+B(x+\widetilde{x})/2+c$, where
$Q(x,\widetilde{x})$ is the symmetric bilinear form corresponding to the
quadratic form $Q(x)$. When applied to integrable systems, Kahan discretization
preserves integrability much more frequently than one would expect a priori,
however not always. We show that in some cases where the original recipe fails
to preserve integrability, one can adjust coefficients of the Kahan
discretization to ensure its integrability.
