Geometric and integrability properties of Kahan's method: the preservation of certain quadratic integrals

Given a quadratic vector field on 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 modified integral. Building on this result, we present a family of integrable quadratic vector fields (including the Euler top) whose Kahan discretization is a family of integrable maps. (paper)