= Solution
The <Jacobian matrix> at the origin is
$$
A=\begin{pmatrix}k&-5\\5&k-6\end{pmatrix}.
$$
Its eigenvalues are
$$
\lambda_\pm=k-3\pm4i.
$$
Thus, when $k>3$, the origin is a hyperbolic repelling spiral. By part (ii), every forward trajectory is eventually trapped in a compact disc. For $k<10$ the hypothesis says that the origin is the only equilibrium; choose any nonstationary trajectory sufficiently close to it. Its <omega-limit set> is nonempty and compact, and it cannot contain the repelling origin. The <Poincaré-Bendixson theorem> then says that this omega-limit set is a periodic orbit. Combining the restrictions gives
$$
\boxed{3<k<10}.
$$
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