By Vinberg theory any homogeneous convex cone
V
\mathscr {V}
may be realised as the cone of positive Hermitian matrices in a
T
T
-algebra of generalised matrices. The level hypersurfaces
V
q
⊂
V
\mathscr {V}_{q} \subset \mathscr {V}
of homogeneous cubic polynomials
q
q
with positive definite Hessian (symmetric) form
g
(
q
)
≔
−
Hess
(
log
(
q
)
)
|
T
V
q
g(q) ≔- \operatorname {Hess}(\log (q))|_{T \mathscr {V}_q}
are the special real manifolds. Such manifolds occur as scalar manifolds of the vector multiplets in
N
=
2
N=2
,
D
=
5
D=5
supergravity and, through the
r
r
-map, correspond to Kähler scalar manifolds in
N
=
2
N = 2
D
=
4
D = 4
supergravity. We offer a simplified exposition of the Vinberg theory in terms of
Nil
\operatorname {Nil}
-algebras (= the subalgebras of upper triangular matrices in Vinberg
T
T
-algebras) and we use it to describe all rational functions on a special Vinberg cone that are
G
0
G_0
- or
G
′
G’
- invariant, where
G
0
G_0
is the unimodular subgroup of the solvable group
G
G
acting simply transitively on the cone, and
G
′
G’
is the unipotent radical of
G
0
G_0
. The results are used to determine
G
0
G_0
- and
G
′
G’
-invariant cubic polynomials
q
q
that are admissible (i.e. such that the hypersurface
V
q
=
{
q
=
1
}
∩
V
\mathscr {V}_q=\{ q=1\}\cap \mathscr {V}
has positive definite Hessian form
g
(
q
)
g(q)
) for rank
2
2
and rank
3
3
special Vinberg cones. We get in this way examples of continuous families of non-homogeneous special real manifolds of cohomogeneity less than or equal to two.