Links and globally completely integrable vector fields on an open 3-manifold

For an oriented link L is an open oriented 3-manifold M, we give a necessary and sufficient condition for the existence of a submersion ϑ:M → R2 whose preimage ϑ−1(0) is L. Also, we prove there always exists a submersion ψ:M → R2 such that the union of the compact components of the preimage ψ−1(0) is L for any link L in M.