Explore the third lecture in a series on infinite-dimensional symplectic geometry, focusing on the quasi-Poisson double case and specific examples involving SU(n) groups. Delve into the mathematical analysis of reductions of integrable master systems on classical doubles of semisimple, connected, and simply connected compact Lie groups. Examine how quotient spaces of internally fused doubles represent moduli spaces of flat principal G-connections on tori with holes. Learn about degenerate integrability inheritance on smooth Poisson quotient components, and understand explicit formulas for reduced Poisson structures and equations of motion using dynamical r-matrices. Study specific applications with small symplectic leaves for G=SU(n), particularly examining compact counterparts of trigonometric Ruijsenaars-Schneider systems. Build upon concepts from previous lectures covering integrable master systems, Heisenberg doubles, and Ruijsenaars-Schneider type many-body systems with spin degrees of freedom.