On the construction of potential and transverse vortex vector fields with lines of zero curvature

Classes of all smooth potential unit vector fields in the domains $R^3$, $R^3\setminus R^0$, $R^3\setminus R^1$ are constructed based on the method of mappings with account taken of the special geometric properties of such fields. Extensions of these classes to classes of all smooth nonunit potential and nonpotential (transverse vortex) fields with straight field lines are found. Their connection with smooth solutions of the corresponding systems of equations is discussed.