  Given a quadratic vector field on \mathbb{R}^n possessing a quadratic first
integral depending on two of the independent variables, we give a constructive
proof that Kahan's discretization method exactly preserves a nearby modifed
integral. Building on this result, we present a family of integrable quadratic
vector fields (including the Euler top) whose Kahan discretization is a novel
10-parameter family of integrable maps.
