The Hill's spherical vortex with a swirling motion is provided by Keith Moffatt in 1969. Earlier discussion of similar problems are provided by William Mitchinson Hicks in 1899. The solution was also discovered by Kelvin H. Pendergast in 1956, in the context of plasma physics, as there exists a direct connection between these fluid flows and plasma physics (see the connection between Hicks equation and Grad–Shafranov equation). The motion  in the axial (or, meridional) plane is described by the Stokes stream function  as before. The azimuthal motion  is given by
where
where  and  are the Bessel functions of the first kind. Unlike the Hills spherical vortex without any swirling motion, the problem here contains an arbitrary parameter . A general class of solutions of the Eulers equation describing propagating three-dimensional vortices without change of shape is provided by Keith Moffatt in 1986.