= Solution
The <gradient> is horizontal when $h_y=0$ and vertical when $h_x=0$. Factoring those <partial derivatives> gives the two <loci>:
$$
\boxed{h_y=0:\quad y=0\quad\text{or}\quad x=\pm\frac{y^2}{2\sqrt2}},
$$
$$
\boxed{h_x=0:\quad x=0\quad\text{or}\quad y=\pm\frac{x^2}{2\sqrt2}}.
$$
Thus each <locus> consists of an axis and two <parabolas>. Their common points are the five <stationary points> found below. At those points the <gradient> is zero and has no direction, so they are included in both algebraic zero-component conditions but excluded if “parallel” is required to mean a nonzero directed <vector>.
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2024/paper-2-q12-loci.png]
{title=Loci of horizontal and vertical gradients}
{description=Blue curves have $h_y=0$, orange curves have $h_x=0$, and black points mark their intersections. At the black points the <gradient> itself vanishes.}
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