= Solution
The $z$ direction is fixed, giving the <eigenvalue> one. The planar block has <characteristic polynomial>
$$
(\cos\phi-\lambda)^2+\sin^2\phi=\lambda^2-2\lambda\cos\phi+1.
$$
Thus the full list of <eigenvalues> is
$$
\boxed{1,\quad e^{i\phi},\quad e^{-i\phi}.}
$$
The last two are real precisely when $\sin\phi=0$, so all three are real exactly for $\boxed{\phi\in\pi\mathbb Z}$. At even multiples of $\pi$ the matrix is the identity and every vector is fixed. At odd multiples it is a half-turn: the axial <eigenvalue> remains $+1$, while the whole perpendicular plane has <eigenvalue> $-1$.
For other angles, the only real <eigenvector> direction is the rotation axis; the perpendicular plane is rotated rather than preserving an individual real direction. Its complex <eigenvectors> can be taken as $(1,-i,0)^T$ and $(1,i,0)^T$ for $e^{i\phi}$ and $e^{-i\phi}$, respectively. These represent the two <complex conjugate> planar eigenvalues.
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