I'm looking to classify the scalar fields $f : \mathbb{R}^3\to\mathbb{R}$ that satisfy $\nabla f\cdot{\bf A} = 0$ for some ${\bf A}\in\mathbb{R}^3$. Since this is a plane passing through the origin with normal vector $\bf A$, I thought I could write $$ \nabla f = \lambda_1{\bf p} + \lambda_2{\bf q} $$ where ${\bf p}\cdot{\bf q} = 0$ and ${\bf p}\times{\bf q} = {\bf A}$, but I wasn't able to figure out any other information about $f$ from this. I also considered the idea that the gradient $\nabla f$ is perpendicular to the level surfaces of $f$, and since $\nabla f$ lies on a plane, maybe this says something about the level surfaces of $f$ (and so about $f$ in turn)? I couldn't take this idea any further however either.
Does anyone have any ideas? Have these types of scalar fields been studied before?