Authors: Alfred Ramani ¹; Basil Grammaticos ¹; Ralph Willox ²; Adrian Carstea ³

• 1 Laboratoire de Physique des 2 Infinis Irène Joliot-Curie [IJCLab]
• 2 University of Tokyo [Tokyo] = Tōkyō teikoku daigaku [UTokyo]
• 3 Horia Hulubei National Institute of Physics and Nuclear Engineering

We derive a class of discrete Painlevé equations associated with the affine Weyl group E$_7^{(1)}$. The method used is the deautonomisation of a QRT mapping belonging to the canonical form VI (according to the classification of said mappings). An equation of such a form was the first instance of a symmetric -- in QRT parlance -- discrete analogue of the Painlevé VI equation. In this paper we present an exhaustive derivation of all the discrete Painlevé equations of this class. This is made possible thanks to previous studies that established the proper lengths of singularity patterns that are compatible with integrablity, and which were already successfully applied to the study of discrete Painlevé equations associated to the affine Weyl group E$_8^{(1)}$. Given that, from the latter, one can obtain by degeneration the equations related to E$_7^{(1)}$, we decided to link the results of the present study to those of the aforementioned ones. It turns out that a bridge from E$_8^{(1)}$ to E$_7^{(1)}$ exists in almost all cases, with one exception where, while in the former case a discrete Painlevé equation does exist, in the latter we find a mapping with only periodic coefficients, devoid of secular dependence.