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

Given a quadratic vector field on ##IMG##
[http://ej.iop.org/images/1751-8121/52/6/065201/aaafb1eieqn001.gif] 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.

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