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We find a novel one-parameter family of integrable quadratic Cremona maps ofthe plane preserving a pencil of curves of degree 6 and of genus 1. They turnout to serve as Kahan-type discretizations of a novel family of quadraticvector fields possessing a polynomial integral of degree 6 whose level curvesare of genus 1, as well. These vector fields are non-homogeneousgeneralizations of reduced Nahm systems for magnetic monopoles with icosahedralsymmetry, introduced by Hitchin, Manton and Murray. The straightforward Kahandiscretization of these novel non-homogeneous systems is non-integrable.However, this drawback is repaired by introducing adjustments of order$O(\epsilon^2)$ in the coefficients of the discretization, where $\epsilon$ isthe stepsize.

How one can repair non-integrable Kahan discretizations. II. A planar system with invariant curves of degree 6