  We study the Hull-Strominger system and the Anomaly flow on a special class
of 2-step solvmanifolds, namely the class of almost-abelian Lie groups. In this
setting, we characterize the existence of invariant solutions to the
Hull-Strominger system with respect to the family of Gauduchon connections in
the anomaly cancellation equation. Then, motivated by the results on the
Anomaly flow, we investigate the flow of invariant metrics in our setting,
proving that it always reduces to a flow of a special form. Finally, under an
extra assumption on the initial metrics, we show that the flow is immortal and,
when the slope parameter is zero, it always converges to a K\"ahler metric
