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We introduce and study a new family of commuting Baxter operators in theRuijsenaars hyperbolic system, different from that considered by us earlier.Using a degeneration of Rains integral identity we verify the commutativitybetween the two families of Baxter operators and explore this fact for theproof of the coupling constant symmetry of the wave function. We also establisha connection between new Baxter operators and Noumi-Sano difference operators.

Baxter operators in Ruijsenaars hyperbolic system IV. Coupling constant reflection symmetry