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We construct a Lax pair for the $E^{(1)}_6 $ $q$-Painlev\'e system from firstprinciples by employing the general theory of semi-classical orthogonalpolynomial systems characterised by divided-difference operators on discrete,quadratic lattices [arXiv:1204.2328]. Our study treats one special case of suchlattices - the $q$-linear lattice - through a natural generalisation of the big$q$-Jacobi weight. As a by-product of our construction we derive the coupledfirst-order $q$-difference equations for the $E^{(1)}_6 $ $q$-Painlev\'esystem, thus verifying our identification. Finally we establish thecorrespondences of our result with the Lax pairs given earlier and separatelyby Sakai and Yamada, through explicit transformations.

Construction of a Lax Pair for the $E_6^{(1)}$ $q$-Painlev\'e System