Lax pairs of discrete Painlevé equations: $(A_2+A_1)^{(1)}$ case

In this paper, we provide a comprehensive method for constructing Lax pairs of discrete Painlev\'e equations. In particular, we consider the $A_5^{(1)}$-surface $q$-Painlev\'e system which has the affine Weyl group symmetry of type $(A_2+A_1)^{(1)}$ and show two new Lax pairs.