We study the 2 × 2 associated linear system (Lax pair) of the q-analogue of the Second and Third Painlevé equation (qPⅡ and qPⅢ).We show that in the cases of rational special solutions for qPⅡ and qPⅢ, the corresponding associated linear problem can be solved exactly in terms of q-Gamma functions.We discuss what this means for the problem of completeness.