Here we have . Integration of  gives  and the three equations are
The first equation is the Hicks equation. Marris and Aswani (1977) showed that the only possible solution is  and the remaining equations reduce to
A simple set of solutions to the above equation is
 represents a flow due to two opposing rotational stream on a parabolic surface,  represents rotational flow on a plane wall,  represents a flow ellipsoidal vortex (special case – Hill's spherical vortex),  represents a type of toroidal vortex etc.
The homogeneous solution for  as shown by Berker
where  are the Bessel function of the first kind and Bessel function of the second kind respectively. A special case of the above solution is Poiseuille flow for cylindrical geometry with transpiration velocities on the walls. Chia-Shun Yih found a solution in 1958 for Poiseuille flow into a sink when .