jats:titleAbstract</jats:title>jats:pWe find a novel one-parameter family of integrable quadratic Cremona maps of the plane preserving a pencil of curves of degree 6 and of genus 1. They turn out to serve as Kahan-type discretizations of a novel family of quadratic vector fields possessing a polynomial integral of degree 6 whose level curves are of genus 1, as well. These vector fields are non-homogeneous generalizations of reduced Nahm systems for magnetic monopoles with icosahedral symmetry, introduced by Hitchin, Manton and Murray. The straightforward Kahan discretization of these novel non-homogeneous systems is non-integrable. However, this drawback is repaired by introducing adjustments of order jats:inline-formulajats:alternativesjats:tex-math$$O(\epsilon ^2)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:miO</mml:mi> mml:mo(</mml:mo> mml:msup mml:miϵ</mml:mi> mml:mn2</mml:mn> </mml:msup> mml:mo)</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> in the coefficients of the discretization, where jats:inline-formulajats:alternativesjats:tex-math$$\epsilon $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:miϵ</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> is the stepsize.</jats:p>