Discrete Painlev\'e equations from singularity patterns: the asymmetric trihomographic case

We derive the discrete Painlev\'e equations associated to the affine Weyl group E$_8^{(1)}$ that can be represented by an (in the QRT sense) "asymmetric" trihomographic system. The method used in this paper is based on singularity confinement. We start by obtaining all possible singularity patterns for a general asymmetric trihomographic system and discard those patterns which cannot lead to confined singularities. Working with the remaining ones we implement the confinement conditions and derive the corresponding discrete Painlev\'e equations, which involve two variables. By eliminating either of these variables we obtain a "symmetric" equation. Examining all these equations of a single variable, we find that they coincide exactly with those derived in previous works of ours, thereby establishing the completeness of our results.