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By Vinberg theory any homogeneous convex cone $\mathcal V$ may be realized asthe cone of positive Hermitian matrices in a $T$-algebra of generalisedmatrices. The level hypersurfaces $\mathcal V_{q} \subset \mathcal V$ ofhomogeneous cubic polynomials $q$ with positive definite Hessian (symmetric)form $g_q := - \operatorname{Hess}(\log(q))|_{T \mathcal V_q}$ are the {\itspecial real manifolds}. Such manifolds occur as scalar manifolds of the vectormultiplets in $N=2$, $D=5$ supergravity and, through the $r$-map, correspond toK\"ahler scalar manifolds in $N = 2$ $D = 4$ supergravity. We offer asimplified exposition of the Vinberg theory in terms of$\operatorname{Nil}$-algebras (= the subalgebras of upper triangular matricesin Vinberg $T$-algebras) and we use it to describe all rational functions on aspecial Vinberg cone that are $G_0$- or $G'$- invariant, where $G_0$ is theunimodular subgroup of the solvable group $G$ acting simply transitively on thecone, and $G'$ is the unipotent radical of $G_0$. The results are used todetermine $G_0$- and $G'$-invariant cubic polynomials $q$ that are {\itadmissible} (i.e. such that the hypersurface $ \mathcal V_q=\{ q=1\}\cap\mathcal V $ has positive definite Hessian form $g_q$) for rank $2$ and rank$3$ special Vinberg cones. We get in this way examples of continuous familiesof non-homogeneous special real manifolds of cohomogeneity less than or equalto two.

Special Vinberg cones, invariant admissible cubics and special real manifolds