A Lax formulation of a generalized $q$-Garnier system

Recently, a birational representation of an extended affine Weyl group of type $A_{mn-1}^{(1)}\times A_{m-1}^{(1)}\times A_{m-1}^{(1)}$ was proposed with the aid of a cluster mutation. In this article we formulate this representation in a framework of a system of $q$-difference equations with $mn\times mn$ matrices. This formulation is called a Lax form and is used to derive a generalization of the $q$-Garnier system.