The symplectic leaves of the Tyurin-Bottacin Poisson structure on stable pure dimension one sheaves on a Poisson surface are the fibers of the map sending a sheaf to the intersection of its support curve with the degeneracy divisor.
Mukai, Symplectic structure of the moduli space of sheaves on an abelian or K3 surface, Invent
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Poisson structure on the moduli spaces of sheaves of pure dimension one on a surface
The symplectic leaves of the Tyurin-Bottacin Poisson structure on stable pure dimension one sheaves on a Poisson surface are the fibers of the map sending a sheaf to the intersection of its support curve with the degeneracy divisor.