Geometrical meaning of the curl operation when A dot curl A does not equal 0

We examine the geometrical interpretation of the vector operation curl for vector fields A such that A dot curl A does not equal 0. It is not possible to construct even locally a surface normal to field lines of A at all points; an explicit geometrical demonstration of this fact is given. Such fields do not admit the construction of orthogonal curvilinear coordinates in which one set of coordinate curves lies along field lines of A.