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

$\mathrm{GL}(2)$-structures in dimension four, $H$-flatness and integrability