Full-parameter discrete Painlevé systems from non-translational Cremona isometries

Since the classification of discrete Painlevé equations in terms of rational surfaces, there has been much interest in the range of integrable equations arising from each of the 22 surface types in Sakai's list (Sakai 2001 Commun. Math. Phys. 220 165–229). For all but the most degenerate type in the list, the surfaces come in families which admit affine Weyl groups of symmetries, translation elements of which define discrete Painlevé equations with the same number of parameters as their family of surfaces. While non-translation elements of the symmetry group have been observed to correspond to discrete systems of Painlevé-type through projective reduction, the resulting equations have fewer than the maximal number of free parameters corresponding to their surface type. We show that equations with the full number of free parameters can be constructed from non-translation elements of infinite order in the symmetry group, constructing several examples and demonstrating their integrability. This is prompted by the study of a previously proposed discrete Painlevé equation related to a special class of discrete analogues of surfaces of constant negative Gaussian curvature (Hoffmann 1999 Oxford Lect. Ser. Math. Appl. 16 83–96). We obtain a full-parameter generalisation of this equation from the Cremona action of a non-translation element of the extended affine Weyl group on a family of generic -surfaces. (paper)