Non-local integrals of motion for deformed $W$-algebras of types $g=A_l,D_l,E_{6,7,8}$

We present an infinite set of non-local integrals of motion for deformed $W$-algebras of types $A_l,D_l$, and $E_{6,7,8}$. 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 $A_l$ and $D_l$ by a direct calculation. In the case of $E_{6,7,8}$ it is a conjecture.