Radial dilation tangency to homogeneous zero sets ($H(\lambda x)={\lambda }^{d}H(x)⟹x\cdot \nabla H=dH$)

Differentiate the displayed homogeneity relation in $\lambda$ at $1$. On every regular point of $H=0$, the radial vector field $x$ is therefore perpendicular to the normal gradient and tangent to the level surface. Its flow $x(t)=e^{t}x(0)$ preserves that zero set. At a singular vertex this flow formulation remains meaningful even if there is no tangent plane.