Extensions of flat partial connections

It is proved that any flat partial del(F) connection VF on a line bundle L over a manifold M, where F is a complex integrable subbundle of (TM)(C) admits an extension to a connection del on L. The kernel of the curvature form omega of del is investigated. These results are applied to description of the Bohr-Sommerfeld subset in geometric quantization theory and quantization of the motion in the potential lambda(2)chi(2) + mu(2)/chi(2).