= Solution
A <left eigenvector> satisfies $(M-\lambda I)^T\mathbf x=0$. Here $M$ is a <symmetric matrix>, so its left and right <eigenspaces> have the same components. The <characteristic polynomial> factors as
$$
\det(M-\lambda I)=(1-\lambda)\bigl[(1-\lambda)^2-4\bigr].
$$
The <eigenvalues> and all corresponding nonzero row <eigenvectors> are
$$
\boxed{\begin{array}{c|c}
\lambda&\mathbf x^T\\\hline
3&c(1,1,0)\\
-1&c(1,-1,0)\\
1&c(0,0,1)
\end{array}\qquad c\ne0}.
$$
For example, in the $xy$ block, equal components give the eigenvalue $3$ and opposite components give $-1$, while the $z$ coordinate is unaffected.
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