= Solution
Expand the given translating-sphere <Stokes flow> into its <Stokeslet> and <potential dipole> parts:
$$
\mathbf u=\frac{3a}{4}\left(\frac{\mathbf U}{r}+\frac{(\mathbf U\cdot\mathbf x)\mathbf x}{r^3}\right)
+\frac{a^3}{4}\left(\frac{\mathbf U}{r^3}-\frac{3(\mathbf U\cdot\mathbf x)\mathbf x}{r^5}\right).
$$
The second parenthesis is $\nabla[(\mathbf U\cdot\mathbf x)/r^3]$ and has zero curl outside the sphere. For the first,
$$
\nabla\times\frac{\mathbf U}{r}=\frac{\mathbf U\times\mathbf x}{r^3},\qquad
\nabla\times\frac{(\mathbf U\cdot\mathbf x)\mathbf x}{r^3}=\frac{\mathbf U\times\mathbf x}{r^3}.
$$
Thus the <vorticity of translating-sphere Stokes flow> is
$$
\boxed{\boldsymbol\omega=\nabla\times\mathbf u=\frac{3a}{2}\frac{\mathbf U\times\mathbf x}{r^3}.}
$$
The result is independent of the irrotational finite-size dipole correction. It holds in the exterior $r>a$; the singular solid harmonics are used only outside their source.
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