  We study five dimensional geometries associated with the 5-dimensional
irreducible representation of GL(2,R). These are special Weyl geometries in
signature (3,2) having the structure group reduced from CO(3,2) to GL(2,R). The
reduction is obtained by means of a conformal class of totally symmetric
3-tensors. Among all GL(2,R) geometries we distinguish a subclass which we term
`nearly integrable GL(2,R)geometries'. These define a unique gl(2,R) connection
which has totally skew symmetric torsion. This torsion splits onto the GL(2,R)
irreducible components having respective dimensions 3 and 7.
  We prove that on the solution space of every 5th order ODE satisfying certain
three nonlinear differential conditions there exists a nearly integrable
GL(2,R) geometry such that the skew symmetric torsion of its unique gl(2,R)
connection is very special. In contrast to an arbitrary nearly integrable
GL(2,R) geometry, it belongs to the 3-dimensional irreducible representation of
GL(2,R). The conditions for the existence of the structure are lower order
equivalents of the Doubrov-Wilczynski conditions found recently by Boris
Doubrov [7].
  We provide nontrivial examples of 5th order ODEs satisfying the three
nonlinear differential conditions, which in turn provides examples of
inhomogeneous GL(2,R) geometries in dimension five, with torsion in R^3. We
also outline the theory and the basic properties of GL(2,R) geometries
associated with n-dimensional irreducible representations of GL(2,R) in 5<n<10.
In particular we give conditions for an n-th order ODE to possess this geometry
on its solution space.
