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Since the classification of discrete Painlev\'e equations in terms ofrational surfaces, there has been much interest in the range of integrableequations arising from each of the 22 surface types in Sakai's list. For allbut the most degenerate type in the list, the surfaces come in families whichadmit affine Weyl groups of symmetries. Translation elements of this symmetrygroup define discrete Painlev\'e equations with the same number of parametersas their family of surfaces. While non-translation elements of the symmetrygroup have been observed to correspond to discrete systems of Painlev\'e-typethrough projective reduction, these have fewer than the maximal number of freeparameters corresponding to their surface type. We show that differenceequations with the full number of free parameters can be constructed fromnon-translation elements of infinite order in the symmetry group, constructingseveral examples and demonstrating their integrability. This is prompted by thestudy of a previously proposed discrete Painlev\'e equation related to aspecial class of discrete analogues of surfaces of constant negative Gaussiancurvature, which we generalise to a full-parameter integrable differenceequation, given by the Cremona action of a non-translation element of theextended affine Weyl group $\widetilde{W}(D_4^{(1)})$ on a family of generic$D_4^{(1)}$- surfaces.

Full-parameter discrete Painlev\'e systems from non-translational Cremona isometries