  We study invariant solutions to the Positive Hermitian Curvature Flow,
introduced by Ustinovskiy, on complex Lie groups. We show in particular that
the canonical scale-static metrics on the special linear groups, arising from
the Killing form, are dynamically unstable. This disproves a conjecture of
Ustinovskiy. We also construct certain perfect Lie groups that admit at least
two distinct invariant solitons for the flow, only one of which is algebraic.
This is the second known example of a geometric flow with non-algebraic,
homogeneous solitons. The first being the G2-Laplacian flow.
