= Solution
Let $M_{ij}=\int_Sx_ix_j\,dS$. A <rotation matrix> $Q$ maps the unit sphere and its surface measure to themselves. Changing variables therefore gives $Q_{ip}Q_{jq}M_{pq}=M_{ij}$, so $M$ is an <isotropic second-rank tensor>. Reflection symmetry also directly makes its off-diagonal entries zero, while interchange of coordinates makes its three diagonal entries equal. Its <trace> is
$$
\sum_i M_{ii}=\int_S|x|^2\,dS=4\pi,
$$
hence $M_{ij}=(4\pi/3)\delta_{ij}$. Oddness gives $\int_Sx_i\,dS=0$. Expanding the defining product for $T$ now gives
$$
\boxed{T_{ij}(y)=\frac{4\pi}{3}\delta_{ij}+4\pi y_iy_j,
\qquad\lambda=\frac{4\pi}{3},\quad\mu=4\pi.}
$$
The action on a vector is $Tv=\lambda v+\mu y(y\cdot v)$. For $y\ne0$, its <eigenvalues> and <eigenspaces> are therefore
$$
\boxed{\frac{4\pi}{3}+4\pi|y|^2\text{ on }\operatorname{span}\{y\},
\qquad\frac{4\pi}{3}\text{ on }y^\perp.}
$$
The latter has multiplicity two. If $y=0$, the single <eigenvalue> $4\pi/3$ has multiplicity three and every vector is an <eigenvector> apart from the excluded zero vector.
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