Modeling egg shaped stars I am well aware of one-dimensional stellar models:

The simplest commonly used model of stellar structure is the spherically symmetric quasi-static model, which assumes that a star is in a steady state and that it is spherically symmetric.  It contains four basic first-order differential equations: two represent how matter and pressure vary with radius; two represent how temperature and luminosity vary with radius.

But what if we moved from spherical symmetry to cylindrical symmetry? Did somebody already set up all equations and solve them for general rotational symmetric ellipsoid?
What changes, if we would assume a lemon-shaped or (most interestingly) an egg-shaped star?
What would be the (intutive) results of such a stellar model? I am sure, somebody solved the equations already and I am just missing the appropriate search terms.
References

The mathematics of egg shape gives a brief mathematical background about one of my favorite mathematical objects

Cylindrical symmetry is not as hypothetical as it might sound:

Ashley Strickland wrote for CNN about &quot;Unusual tear-drop shaped, half-pulsating star discovered by amateur astronomers&quot;
WASP-12b is is reviewed by NASA as An egg-shaped planet.

The pre-print by E.C. &amp; L.V. Nolan On isotropic cylindrically symmetric stellar models seems to cover the topic, but is not too intuitive.
Related

Can a donut-shape planet or star be formed?
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.