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At the turn of the century, Etingof and Sevostyanov independently constructeda family of quantum integrable systems, quantizing the open Toda chainassociated to a simple Lie group $G$. The elements of this family areparameterized by Coxeter words of the corresponding Weyl group. Twenty yearslater, in the works of Finkelberg, Gonin, and Tsymbaliuk, this was generalizedto a family of quantum Toda chains parameterized by pairs of Coxeter words. Inthis paper, we show that this family is actually a single cluster integrablesystem written in different clusters associated to cyclic double Coxeter words.Furthermore, if we restrict the action of Hamiltonians to its positiverepresentation, these systems become unitary equivalent.

Lax Matrices & Clusters for Type A & C Q-Deformed Open Toda Chain