= Solution
Take the Dulac function $B=1$. For
$$
Q=5x^2-6xy+5y^2,
$$
direct differentiation gives
$$
\begin{aligned}
\frac{\partial\dot x}{\partial x}
+\frac{\partial\dot y}{\partial y}
&=2(k-3)-2(27x^2-26xy+27y^2).
\end{aligned}
$$
The quadratic form in parentheses has matrix
$$
\begin{pmatrix}27&-13\\-13&27\end{pmatrix},
$$
whose <eigenvalues> are $14$ and $40$, so it is <positive-definite matrix>[positive definite]. For $0<k\leq3$ the divergence is nonpositive and vanishes only at the origin when $k=3$. The <Bendixson-Dulac criterion> therefore gives
$$
\boxed{\text{there is no periodic orbit for }0<k\leq3}.
$$
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