= Solution
Write the angular dependence as $f(\phi)=\sum_{n\ge1}a_n\sin n\phi+\sum_{n\ge0}b_n\cos n\phi$, so $u'=\sin\theta f(\phi)e_\phi$. In the <spherical coordinate system>, $n=e_r$ and $e_r\times e_\phi=-e_\theta$, with
$$
e_\theta=(\cos\theta\cos\phi,\cos\theta\sin\phi,-\sin\theta),\qquad
dS=a^2\sin\theta\,d\theta\,d\phi.
$$
Thus the integral in the <torque-free rotation of a spherical squirmer> has components
$$
I_x=-a^2\left(\int_0^\pi\sin^2\theta\cos\theta d\theta\right)
\left(\int_0^{2\pi}f(\phi)\cos\phi d\phi\right)=0,
$$
$$
I_y=-a^2\left(\int_0^\pi\sin^2\theta\cos\theta d\theta\right)
\left(\int_0^{2\pi}f(\phi)\sin\phi d\phi\right)=0.
$$
The vanishing follows from north-south cancellation, including the potentially relevant first azimuthal harmonics. In the $z$ direction, <orthogonality of complex exponentials> gives
$$
I_z=a^2\int_0^\pi\sin^3\theta d\theta\int_0^{2\pi}f(\phi)d\phi
=a^2\frac43(2\pi b_0)=\frac{8\pi a^2}{3}b_0.
$$
Consequently every rotational component is determined:
$$
\boxed{\Omega_x=0,\qquad\Omega_y=0,\qquad\Omega_z=-\frac{b_0}{a}.}
$$
This assumes an integrable prescribed angular function, with the Fourier series convergent sufficiently to justify the surface integrals; a finite Fourier sum is enough for the construction below. Other harmonics can generate exterior flows even though they contribute no net rotation.
For <flow-free rotation of a spherical squirmer>, choose $b_0\ne0$ and set every $a_n$ and every $b_n$ with $n\ge1$ to zero. Then $u'=b_0\sin\theta e_\phi$, while the rigid-rotation velocity is
$$
\Omega\times an=-b_0\sin\theta e_\phi=-u'.
$$
The total laboratory-frame boundary velocity is therefore zero everywhere. With fluid at rest at infinity, <Uniqueness of Stokes flow> gives
$$
\boxed{u(r)=0\quad\text{throughout the exterior fluid, although }\Omega_z=-b_0/a\ne0.}
$$
A constant pressure supplies zero resultant force and torque, so this is consistent with free motion. The imposed surface actuation counter-rotates relative to the material sphere; no claim is being made that ordinary no-slip rigid rotation produces zero flow.
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