The dynamical Alekseevskii conjecture in dimension five

We prove the dynamical Alekseevski conjecture in dimension five. We also provide a detailed analysis of the homogeneous Ricci flows on $SO(3)⋉{\mathbb{R}}^3\mathrm{/}SO(2)$ and $SL(2,\mathbb{C})\mathrm{/}U(1)$.