On irrotational vector fields with vector lines located on a given surface

Irrotational vector fields are considered on the surface given by the equation $a=z+\alpha(x,y,t)=0$. The conditions under which the vector lines of such fields are located on this surface are studied. Sufficient conditions for the existence of a harmonic vector field with such vector lines are obtained. An overdetermined system of partial differential equations is studied, the solution of which provides a harmonic field, the vector lines of which lie on a given surface of the considered type. The equation of the surface is written, for which it is possible to find a harmonic vector field with vector lines located on this surface. It is shown that for any surfaces of the type under consideration, one can find irrotational nonharmonic vector fields with vector lines located on a given surface. A number of surfaces are given for which harmonic or nonharmonic irrotational vector fields with vector lines located on these surfaces are obtained. Navier - Stokes equations for a viscous incompressible fluid in nondimensional form are considered. For this system, under the assumption that the velocity field is potential, a particular solution is written that ensures the location of the vector lines of the velocity field on a paraboloid of revolution.