We present an infinite set of non-local integrals of motion for the deformed W-algebra Wₐ, ₓ (g) associated with the affine Lie algebras g=Aₗ^ (1), Dₗ^ (1) and E₆, ₇, ₈^ (1). They can be regarded as a two-parameter deformation of trace of the monodromy matrix of the g-KdV theory. Commutativity of the non-local integrals of motion is shown in the case of g=Aₗ^ (1) and Dₗ^ (1) by a direct calculation. In the case of g=E₆, ₇, ₈^ (1) it is a conjecture.
Jimbo et al. (Sun,) studied this question.