steps: find a suitable group $\mathcal{G}$ of rational maps; apply the elements of the group to the functional equation and form a telescoping sum; and (ultimately) represent the generating function of a model as the positive series extraction of an explicit rational function. Bousquet-M[é]{}lou and Mishna  illustrate the applicability in the case of lattice walks, and it has been adapted to several dimensions [@BoBoKaMe14].

### The group $\mathcal{G}$

For any $d$-dimensional model, we define the group $\mathcal{G}$ of $2^d$ rational maps by $$\mathcal{G} := \left\{(z_1,\dots,z_d) \mapsto (z_1^{i_1},\dots,z_d^{i_d}) : (i_1,\dots,i_d) \in \{-1,1\}^d \right\}.$$ Given $\sigma \in \mathcal{G}$, we can consider $\sigma$ as a map on $\mathbb{Q}[z_1,{\ensuremath{\overline{z}}}_1,\dots,z_d,{\ensuremath{\overline{z}}}_d][\![t]\!]$ through the group action defined by . Due to the symmetry of the step set across each axis, one can verify that $\sigma(S({\ensuremath{\mathbf{z}}}))=S(\sigma({\ensuremath{\mathbf{z}}}))=S({\ensuremath{\mathbf{z}}})$ always holds. The fact that this group does not depend on the step set of the model – only on the dimension $d$ – is crucial to obtaining the general results here. When $d$ equals two, the group $\mathcal{G}$ matches the group used by [@FaIaMa99] and [@BoMi10]. As we will see in Section \[sec:Weyl\], $\mathcal{G}$ corresponds to the Weyl group of the Weyl chamber $A_1^d$, where the step set ${\ensuremath{\mathcal{S}}}$ can be 