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We give a characterisation of Atiyah's and Hitchin's transverse Hilbertschemes of points on a symplectic surface in terms of bi-Poisson structures.Furthermore, we describe the geometry of hyperk\"ahler manifolds arising fromthe transverse Hilbert scheme construction, with particular attention paid tothe monopole moduli spaces.

Transverse Hilbert schemes, bi-Hamiltonian systems, and hyperk\"ahler geometry