The gradient is horizontal when and vertical when . Factoring those partial derivatives gives the two loci:
(1)
(2)
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.
Figure 1.
Loci of horizontal and vertical gradients
. Blue curves have , orange curves have , and black points mark their intersections. At the black points the gradient itself vanishes. Details.

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