Additive actions on projective hypersurfaces with a finite number of orbits

An induced additive action on a projective variety $X\subseteq {\mathbb{P}}^n$ is a regular action of the group ${\mathbb{G}}_a^m$ on $X$ with an open orbit, which can be extended to a regular action on the ambient projective space ${\mathbb{P}}^n$. In this work, we classify all projective hypersurfaces admitting an induced additive action with a finite number of orbits.