= Solution
Write the first-order surface <velocity> as $\mathbf u_1=\mathbf U_1+\boldsymbol\Omega_1\times\mathbf x+\mathbf g$, where the previous derivative gives
$$
\mathbf g=\frac{3f}{2a}(\mathbf I-\mathbf n\mathbf n)\mathbf U_0.
$$
As a test field, choose a sphere translating with arbitrary <velocity> $\widehat{\mathbf V}$. Its surface <velocity> is $\widehat{\mathbf V}$ and its <traction> is $\widehat{\boldsymbol\sigma}\mathbf n=-3\mu\widehat{\mathbf V}/(2a)$. The <Lorentz reciprocal theorem for Stokes flow> gives
$$
\int_{r=a}\mathbf u_1\cdot\widehat{\boldsymbol\sigma}\mathbf n\,dS
=\int_{r=a}\widehat{\mathbf V}\cdot\boldsymbol\sigma_1\mathbf n\,dS=0.
$$
Terms at infinity vanish because both <velocities> decay. Since $\int\mathbf n\,dS=0$, the rotation term has zero surface average. The arbitrary test <velocity> therefore gives the <first-order mobility of a nearly spherical particle>:
$$
\boxed{\mathbf U_1=-\frac1{4\pi a^2}\int_{r=a}\mathbf g\,dS
=\frac{3}{8\pi a^3}\int_{r=a}f(\mathbf n)(\mathbf n\mathbf n-\mathbf I)\mathbf U_0\,dS.}
$$
For the ellipsoidal perturbation, the second spherical moment gives $\int f\,dS=(4\pi a^3/3)\operatorname{tr}\mathbf D=0$. The supplied fourth moment gives, componentwise,
$$
\int f n_i n_j\,dS
=\frac{4\pi a^3}{15}\left(\delta_{ij}\operatorname{tr}\mathbf D+D_{ij}+D_{ji}\right)
=\frac{8\pi a^3}{15}D_{ij}.
$$
Substitution yields
$$
\boxed{\mathbf U_1=\frac15\mathbf D\mathbf U_0=\frac15\mathbf U_0\cdot\mathbf D,}
$$
where the last notation uses symmetry of the <tensor>.
One can also use a rotating-sphere test with surface <traction> $-3\mu\widehat{\boldsymbol\Omega}\times\mathbf n$. Its reciprocal integral, together with zero first-order <torque>, gives
$$
\boldsymbol\Omega_1=-\frac{3}{8\pi a^3}\int_{r=a}\mathbf n\times\mathbf g\,dS
=-\frac{9}{16\pi a^4}\int_{r=a}f(\mathbf n)\mathbf n\times\mathbf U_0\,dS.
$$
Here $f(-\mathbf n)=f(\mathbf n)$, so the last integrand is odd and its integral vanishes. Thus \b[$\boldsymbol\Omega_1=0$]. Inversion symmetry forbids a translation-to-rotation coupling for this centred ellipsoidal perturbation.
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