  We show that torsion-free four-dimensional $\mathrm{GL}(2)$-structures are
flat up to a coframe transformation with a mapping taking values in a certain
subgroup $H\subset\mathrm{SL}(4,\mathbb{R})$ which is isomorphic to a
semidirect product of the three-dimensional continuous Heisenberg group
$H_3(\mathbb{R})$ and the Abelian group $\mathbb{R}$. In addition, we show that
the relevant PDE system is integrable in the sense that it admits a
dispersionless Lax-pair.
