Abstract
The Kahan discretization of the Lotka-Volterra system, associated with any skew-symmetric graph Γ, leads to a family of rational maps, parametrized by the step size. When these maps are Poisson maps with respect to the quadratic Poisson structure of the Lotka-Volterra system, we say that the graph Γ has the Kahan-Poisson property. We show that if Γ is connected, it has the Kahan-Poisson property if and only if it is a cloning of a graph with vertices $1,2,\dots ,n$ , with an arc i → j precisely when i < j, and with all arcs having the same value. We also prove a similar result for augmented graphs, which correspond with deformed Lotka-Volterra systems and show that the obtained Lotka-Volterra systems and their Kahan discretizations are superintegrable as well as Liouville integrable.
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