= Solution
Write the coefficient matrix as $M$. Its <characteristic polynomial> is $\det(rI-M)=r(r-2)(r+2)$. Corresponding <eigenvectors> are
$$
v_-=(-1,1,1)^T,\qquad v_0=(1,1,1)^T,\qquad
v_+=(-1,-1,1)^T,
$$
with <eigenvalues> $-2,0,2$ respectively, as direct multiplication verifies. Distinct <eigenvalues> make these vectors a basis. Expanding a solution as $z_-v_-+z_0v_0+z_+v_+$ decouples the <linear ordinary differential equations> into $\dot z_-=-2z_-$, $\dot z_0=0$, $\dot z_+=2z_+$. Thus \b[the general solution is]
$$
\boxed{\begin{pmatrix}x\\y\\z\end{pmatrix}
=Ae^{-2t}v_-+Bv_0+Ce^{2t}v_+.}
$$
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