  We study the flow of Hermitian metrics governed by the second Chern-Ricci
form on a compact complex manifolds. The flow belongs to the family of
Hermitian curvature flows introduced by Streets and Tian and it was considered
by Lee in order to study compact Hermitian manifolds with almost negative Chern
bisectional curvature. We show a regularity result on compact complex
parallelizable manifolds and we prove that Chern-flat metrics are dynamically
stable.
