EDIT: Note that the object I'm seeking needn't have anything to do with water or actual containers; those are just used to convey the idea.
I'm trying to find a container that, when turned with some constant angular velocity in any direction given by $\phi$ (rotational symmetry), will pour out the same amount of water per time, regardless of how much water is left in it (there must, of course, be some water in it though).
This problem can be reduced to finding a container for which, when looking at the cross-section, the line between the two points that touch both the water surface and the container is of constant length $l$ for any angle with the horizontal, $\theta \in [\theta_0=0,\theta_{max} \leq \frac{\pi}{4}]$, we might hold the container at. (Note that $\theta_{max}$ is just the angle at which all of the water has left the container; the desired container doesn't have to live up to anything after this point. For instance it can be shown that if the cross-section is a parabola, then the all water will have left at $\theta_{max}=\arctan(4) \approx 76^{\circ}$.)
The container directly below does meet some of the criteria.
$\hspace{3.5cm}$$\hspace{3cm}$
If we look at the cross-section (with water in blue), we see that, because it is a quarter circle, $l$ (the radius) is constant throughout.
$\hspace{2.8cm}$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\underset{pouring}{\rightarrow}$
The problem, however, is that it is not rotationally symmetric with $\phi$, i.e. $l$ does not remain constant regardless of the direction we are pouring.
I instead imagine some kind of function $f$ (like the one displayed below)
$\hspace{6.6cm}$
that will show rotational symmetry around the shown $y$-axis, when integrated around said axis (to get the 3D container).
I've tried messing around with it a lot, but I don't really know how to formalize this properly.
Q: Does such a container exist and if so, what is it?
If not, would it be possible to construct it higher dimensions?