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We study wall-crossing for the Beauville-Mukai system of rank three on ageneral genus two K3 surface. We show that such a system is related to theHilbert scheme of ten points on the surface by a sequence of flops, whoseexceptional loci can be described as Brill-Noether loci. We also obtainBrill-Noether type results for sheaves in the Beauville-Mukai system.

Birational geometry of Beauville-Mukai systems I: the rank three and genus two case