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We consider sets of screening operators with fermionic screening currents. Westudy sums of vertex operators which formally commute with the screeningoperators assuming that each vertex operator has rational contractions with allscreening currents with only simple poles. We develop and use the method of$qq$-characters which are combinatorial objects described in terms of deformedCartan matrix. We show that each qq-character gives rise to a sum of vertexoperators commuting with screening operators and describe ways to understandthe sum in the case it is infinite.  We discuss combinatorics of the qq-characters and their relation to theq-characters of representations of quantum groups.  We provide a number of explicit examples of the qq-characters with theemphasis on the case of $D(2,1;\alpha)$. We describe a relationship of theexamples to various integrals of motion.

Combinatorics of vertex operators and deformed $W$-algebra of type D$(2,1;\alpha)$