The authors study embeddings of complex vector bundles, especially line bundles in the complexification of the tangent bundle \(CTM\) of a manifold \(M\) and its implications in the establishment of properties of interest in partial differential equations. The main questions considered are: 1. Let \({\mathcal V}\), be a complex vector bundle \(E\to M\) of rank \(\leq\dim (M)\), is it realizable as a sub-bundle of \(CTM\)? 2. If the complex vector bundle \({\mathcal V}\), \(E\to M\), can be realized as a sub-bundle of \(CTM\), can it be realized as an involutive sub-bundle (that is, if \(X\) and \(Y\) are sections of \({\mathcal V}\), then their Lie bracket \([X,Y]\) is again a section of \({\mathcal V})\)? The Bott's necessary conditon for involutivity is briefly discussed. 3. Embeddings of complex line bundles \({\mathcal V}\) as CR sub-bundles (that is, \({\mathcal V}\) involutive and \({\mathcal V}\cap \overline {cal V}=0)\). The authors include two theorems of Hirzebruch and Hopf on the existence of almost complex structures and two-plane fields in compact 4-manifolds. 4. The construction of a number (an integer) as a measure of the characteristic set of a generically embededded bundle in \(CTM\). 5. The existence of hypo-complex structures (this class of structures is modelled on, but does not coincide with, the complex structures) on an orientable 2-manifold with the construction of all global, hypo-complex structures on such manifolds. 6. The proof that at least for rank 1 sub-bundles of \(CTM\), \(M\) of any dimension, there is no topological obstruction to the existence of hypo-analytic structures.