  We give a birational description of the reduced schemes underlying the
irreducible components of the nilpotent cone and the $\mathbb{C}^\times$-fixed
point locus of length two in the moduli space of Higgs bundles. We show that
fixed point components of type $(n_0,n_1)$ are wobbly, and that they are also
$\mathrm{U}(n_0,n_1)$-wobbly (an \`a priori stronger notion). We analyse their
virtual equivariant multiplicities and Euler pairings with downward flows for
type $(1,\dots, 1)$ fixed points, to find that both invariants fail to fully
detect all wobbly components for partitions other than $(2,1)$.
