The stream function
ψ
\psi
for axisymmetric Stokes flow satisfies the well-known equation
E
4
ψ
=
0
{E^4}\psi = 0
. In spheroidal coordinates the equation
E
2
ψ
=
0
{E^2}\psi = 0
admits separable solutions in the form of products of Gegenbauer functions of the first and second kind, and the general solution is then represented as a series expansion in terms of these eigenfunctions. Unfortunately, this property of separability is not preserved when one seeks solutions of the equation
E
4
ψ
=
0
{E^4}\psi = 0
. The nonseparability of
E
4
ψ
=
0
{E^4}\psi = 0
in spheroidal coordinates has impeded considerably the development of theoretical models involving particle-fluid interactions around spheroidal objects. In the present work the complete solution for
ψ
\psi
in spheroidal coordinates is obtained as follows. First, the generalized 0-eigenspace of the operator
E
2
{E^2}
is investigated and a complete set of generalized eigenfunctions is given in closed form, in terms of products of Gegenbauer functions with mixed order. The general Stokes stream function is then represented as the sum of two functions: one from the 0-eigenspace and one from the generalized 0-eigenspace of the operator
E
2
{E^2}
. A rearrangement of the complete expansion, in such a way that the angular-type dependence enters through the Gegenbauer functions of successive order, leads to some kind of semiseparable solutions, which are given in terms of full series expansions. The proper solution subspace that provides velocity and vorticity fields, which are regular on the axis, is given explicitly. Finally, it is shown how these simple and generalized eigenfunctions reduce to the corresponding spherical eigenfunctions as the focal distance of the spheroidal system tends to zero, in which case the separability is regained. The usefulness of the method is demonstrated by solving the problem of the flow in a fluid cell contained between two confocal spheroidal surfaces with Kuwabara-type boundary conditions.