Is it really necessary for the orbital plane of a satellite to pass through the center of mass of the object around which it's orbiting?

Is it really necessary for the orbital plane of a satellite to pass through the center of mass of the celestial body around which it's orbiting? Does the answer depend on, whether the celestial body is a sphere of radially symmetric density or it's irregularly shaped (for example, a comet or an asteroid)? If there exist cases where the orbital plane doesn't pass through the center of mass, how is Kepler's First Law defined for such cases, since the center of mass of the larger body doesn't lie on the orbital plane of the satellite which contains the foci of the elliptical orbit?

Please Note: The answers for the question Why do satellites orbit around the centre of a planet? did not clarify my above mentioned doubts.

• Let's say we have a point mass $m_0$ present in the vicinity of a giant arbitrarily-shaped planet of mass $M$ and volume $V$. The net gravitational force acting on the point mass due to the planet is given by $\int_V Gm_0 dm \frac{\mathbf{r}}{r^3}$ (where $\mathbf{r}$ is the position vector from $m_0$ to the infinitesimal mass $dm$ : origin is located at $m_0$). Is there a special name for this point : $\int_V dm \frac{\mathbf{r}}{Mr^3}$ ? (Similar to how we call $\int_V dm \frac{\mathbf{r}}{M}$ as the COM of the planet) Nov 27, 2019 at 8:35
• @AjayMohan The point you refer to depends on the position of the test mass $m_0$. It is only if the large object has a mass distribution that is sherically symmetric that the point you refer to coincides with the center of symmetry (which is perforce the center of mass) wherever the test mass $m_0$ is. Otherwise, it does depend on $m_0$'s position ! Nov 27, 2019 at 9:02