The full class ofsmooth axially symmetric longitudinal-vortex unit vector fields

In the paper, two vector fields are constructed by means of transformation method. The first describes the axially symmetric unit solutions (ASUS) of the Gromeka problem to find out vector fields which flow lines coincide in $R^3$ with vortex lines. The second describes the smooth ASUS of the extended in this paper Gromeka problem of finding a vector fields with different vortex properties in adjacent parts of $R^3$.