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Closed form expressions in terms of multi-sums of products have been given in\cite{Tranclosedform, KRQ} of integrals of sine-Gordon, modified Korteweg-deVries and potential Korteweg-de Vries maps obtained as so-called$(p,-1)$-traveling wave reductions of the corresponding partial differenceequations. We prove the involutivity of these integrals with respect torecently found symplectic structures for those maps. The proof is based onexplicit formulae for the Poisson brackets between multi-sums of products.

Involutivity of integrals for sine-Gordon, modified KdV and potential KdV maps