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We first exhibit two compatible Poisson structures on the cotangent bundle ofthe unitary group $\mathrm{U}(n)$ in such a way that the invariant functions ofthe $\mathfrak{u}(n)^*$-valued momenta generate a bi-Hamiltonian hierarchy. Oneof the Poisson structures is the canonical one and the other one arises fromembedding the Heisenberg double of the Poisson-Lie group $\mathrm{U}(n)$ into$T^*\mathrm{U}(n)$, and subsequently extending the embedded Poisson structureto the full cotangent bundle. We then apply Poisson reduction to thebi-Hamiltonian hierarchy on $T^*\mathrm{U}(n)$ using the conjugation action of$\mathrm{U}(n)$, for which the ring of invariant functions is closed under bothPoisson brackets. We demonstrate that the reduced hierarchy belongs to theoverlap of well-known trigonometric spin Sutherland and spinRuijsenaars--Schneider type integrable many-body models, which receive abi-Hamiltonian interpretation via our treatment.

Reduction of a bi-Hamiltonian hierarchy on $T^*\mathrm{U}(n)$ to spin Ruijsenaars--Sutherland models