= Solution
With $\nu=0$, the <characteristic polynomial> factors into $(1-\lambda)[\lambda^2-\lambda-(\mu^2+4)]$, up to the overall sign convention. The <eigenvalues> are $1$ and $(1\pm\sqrt{17+4\mu^2})/2$. Integer values require an odd integer $m$ satisfying $(m-2\mu)(m+2\mu)=17$. The positive integer factorization $17=1\cdot17$ forces $m=9$ and
$$
\boxed{\mu=4.}
$$
The resulting <eigenvalues> and corresponding nonzero <eigenvectors> are
$$
\boxed{\begin{array}{c|c}1&(2,0,-1)^T\\5&(1,2,2)^T\\-4&(2,-5,4)^T\end{array}.}
$$
Any nonzero scalar multiple is equally valid. The factorization also shows that no other positive integer $\mu$ works.
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