  Building on previous work [1 and 2] on the system of bosons at filling factor
$\nu = 1$ we derive the Dirac composite fermion theory for a half-filled Landau
level from first principles and applying Hartree-Fock approach in a preferred
representation. On the basis of the microscopic formulation, in the
long-wavelength limit, we propose a non-commutative field-theoretical
description, which in a commutative limit reproduces the Son's theory, with
additional terms that may be expected on physical grounds. The microscopic
representation of the problem is also used to discuss pairing instabilities of
composite fermions. We find that a presence of a particle-hole symmetry
breaking leads to a weak (BCS) coupling $p$-wave pairing in the lowest Landau
level, and strong coupling $p$-wave pairing in the second Landau level that
occurs in a band with nearly flat dispersion - a third power function of
momentum.
