We study the nilpotent cone in the Mukai system of rank two and genus two. We compute the degrees and multiplicities of its irreducible components and describe their cohomology classes.

Using the techniques of Bayer-Macrì, we determine the walls in the movable cone of the Mukai system of rank two for a general K3 surface S of genus two. We study the (essentially unique) birational map to S [5] and decompose it into a sequence of flops. We give an interpretation of the exceptional loci in terms of Brill-Noether loci.

We study the nilpotent cone in the Mukai system for rank two and genus two. We compute the degrees and multiplicities of its irreducible components and describe their cohomology classes.

Combining theorems of Voisin and Marian, Shen, Yin and Zhao, we compute the dimensions of the orbits under rational equivalence in the Mukai system of rank two and genus two. We produce several examples of algebraically coisotropic and constant cycle subvarieties.