  We study $GL(2)$-structures on differential manifolds. The structures play a
fundamental role in the geometric theory of ordinary differential equations. We
prove that any $GL(2)$-structure on an even dimensional manifold give rise to a
certain almost-complex structure on a bundle over the original manifold.
Further, we exploit a natural notion of integrability for the
$GL(2)$-structures, which is a counterpart of the self-duality for the
4-dimensional conformal structures. We relate the integrability of the
$GL(2)$-structures to the integrability of the almost-complex structures. This
allows to perform a twistor-like construction for the $GL(2)$-geometry.
Moreover, we provide an explicit construction of a canonical connection for any
$GL(2)$-structure.
