  We give a complete list, for $n \leq 6$, of non-isometric
$\mathbb{T}^n$-invariant Kaehler-Einstein manifolds immersed in a finite
dimensional complex projective space endowed with the Fubini-Study metric. This
solves, in the aforementioned case, a classical and long-staying problem
addressed among others by Calabi and Chern.
