Determination of the symmetry group for some QRT roots

We determine the affine Weyl symmetries of some two-dimensional birational maps known as QRT roots arising from Kahan–Hirota–Kimura discretisation of two different reduced Nahm systems. The main finding is that the symmetry types of these discrete systems are subgroups of the Weyl groups for Sakai’s discrete Painlevé equations to which the QRT maps are the autonomous limits.