  We explicitly describe the solution of the G$_2$-Laplacian flow starting from
an extremally Ricci-pinched closed G$_2$-structure on a compact 7-manifold and
we investigate its properties. In particular, we show that the solution exists
for all real times and that it remains extremally Ricci-pinched. This result
holds more generally on any 7-manifold whenever the intrinsic torsion of the
extremally Ricci-pinched G$_2$-structure has constant norm. We also discuss
various examples.
