Modeling egg shaped stars
Diclaimer: This is not (yet) an answer! To attract answers, I decided to start an answer draft which can be expanded by others.
Cylindrical coordinates
Every point in our cylindrical coordinate system is defined by a tuple $(r,\varphi,z)$ where $r$ is the distance from the rotational axis. We also define $Z$ as the height of our solid of revolution, i.e. $0 \leq z \leq Z$. The shape of the body is defined by shape function $s(z)$.
The volume $V$ of the object is then given by $$V= \pi \int_0^Z \left( s(z) \right)^2 {\rm d}z$$
Mass conservation
The mass density $\rho(r,z)$ does not depend on $\varphi$.
to be continued
Specific shape curves
Up to now, all maths has been performed for a general shape function $s(z)$, so let us now look at some specific ones
Egg as rotational body
For an egg with $z$ being the distance from the symmetry axis, we could for instance a formula by Narushin:
$$s(z) = 1.5396 \cdot \frac{B}{Z} \cdot\sqrt{ \sqrt{Z}\cdot z^{\frac{3}{2}}-z^2}$$
In this formula, $B$ is the maximum breadth and $Z$ is the height of the egg.