$$ f(x)=f \left( x \pm \frac{l}{\sqrt{1+\dot{f}^2}} \right) \mp \frac{l}{\sqrt{1+\dot{f}^2}}\dot{f}^2, $$ where $l$ is a constant. How is such a beast even approached? If anyone got intuition for educated guesses, I'm all ears.
EDIT 1: I'm trying to find a container that, when turned with some constant angular velocity in any direction (rotational symmetry), will pour out the same amount of water per time, regardless of how much water is left in it (there must be some water in it though, of course). The expression in the main text is found by looking at a midsection of the container, where the outlines are given by $f(x)$ and $f(-x)$. I then say that the tangent to any point of $f$ is the surface of the water, and then I find the length $l$ between the point of the tangent and the point where the tangent intersects with $f(-x)$ (on the "back" of the container). If $l$ is constant, the flow of water will be as well.
EDIT 2: The question described in the first edit is fleshed out in my more recent question (which sadly doesn't yet have an answer).