= Solution
The <Linearity of Stokes flow> makes every velocity linear in $\mathbf G_1$. The only available isotropic polar vector built from the axial vector $\mathbf G_1$ and separation vector $\mathbf R$ is $\mathbf G_1\times\mathbf R$. <Dimensional analysis> then gives
$$
\mathbf U_i=\alpha_i(R/a)\frac{\mathbf G_1\times\mathbf R}{\mu a^3}.
$$
An angular velocity is axial, so the two independent isotropic possibilities are $\mathbf G_1$ and $(\mathbf G_1\cdot\mathbf R)\mathbf R$. Hence
$$
\boxed{
\boldsymbol\Omega_i
=\frac1{\mu a^3}\left[
\beta_i(R/a)\mathbf G_1
+\gamma_i(R/a)\frac{(\mathbf G_1\cdot\mathbf R)\mathbf R}{R^2}
\right],}
$$
for dimensionless scalar functions $\beta_i,\gamma_i$.
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