= Solution
By the <spectral theorem for real symmetric matrices>, choose an <orthonormal eigenbasis> $\mathbf u_1,\mathbf u_2,\mathbf u_3$ corresponding to $6,\sqrt3,-\sqrt3$, with $\mathbf u_1=(1,1,1)^T/\sqrt3$. Write $P=[\mathbf u_1\ \mathbf u_2\ \mathbf u_3]$ and $\mathbf x=P\boldsymbol\xi$. This <orthogonal change of coordinates> gives
$$
\mathbf x^TM\mathbf x=6\xi_1^2+\sqrt3\xi_2^2-\sqrt3\xi_3^2,\qquad
x_1+x_2+x_3=\sqrt3\xi_1.
$$
The plane constraint is therefore $\xi_1=0$. On that plane the <quadratic form> vanishes exactly when
$$
\xi_2^2-\xi_3^2=(\xi_2-\xi_3)(\xi_2+\xi_3)=0.
$$
Consequently the entire set is
$$
\boxed{\{t(\mathbf u_2+\mathbf u_3):t\in\mathbb R\}
\ \cup\ \{t(\mathbf u_2-\mathbf u_3):t\in\mathbb R\}}.
$$
The two direction <vectors> are nonzero and not parallel, since $\mathbf u_2,\mathbf u_3$ are <linearly independent>. An invertible <orthogonal transformation> maps each full line to a full line. Thus the set consists of precisely two infinite lines through the origin.
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