  We determine the Hausdorff dimension of $k$-multiple points for a symmetric
operator semistable L\'evy process $X=\{X(t), t\in\mathbb{R}_+\}$ in terms of
the eigenvalues of its stability exponent. We also give a necessary and
sufficient condition for the existence of $k$-multiple points. Our results
extend to all $k\geq2$ the recent work [23], where the set of double points $(k
= 2)$ was studied in the symmetric operator stable case.
