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We consider complete toric varieties $X$ such that a maximal unipotentsubgroup $U$ of the automorphism group $\text{Aut}(X)$ acts on $X$ with an openorbit. It turns out that such varieties can be characterized by severalremarkable properties. We study the set of Demazure roots of the correspondingcomplete fan, describe the structure of a maximal unipotent subgroup $U$ in$\text{Aut}(X)$, and find all regular subgroups in $U$ that act on $X$ with anopen orbit.

Radiant toric varieties and unipotent group actions