Spatially inhomogeneous and irrotational geometries admitting Intrinsic
  Conformal Symmetries
  "Diagonal" spatially inhomogeneous (SI) models are introduced under the
assumption of the existence of (proper) intrinsic symmetries and can be seen,
in some sense, complementary to the Szekeres models. The structure of this
class of spacetimes can be regarded as a generalization of the (twist-free)
Locally Rotationally Symmetric (LRS) geometries without any global isometry
containing, however, these models as special cases. We consider geometries
where a six-dimensional algebra $\mathcal{IC}$ of Intrinsic Conformal Vector
Fields (ICVFs) exists acting on a $2-$dimensional (pseudo)-Riemannian manifold.
Its members $\mathbf{X}_{\alpha }$, constituted of 3 Intrinsic Killing Vector
Fields (IKVFs) and 3 \emph{proper} and \emph{gradient} ICVFs, as well as the
specific form of the gravitational field are given explicitly. An interesting
consequence, in contrast with the Szekeres models, is the immediate existence
of \emph{conserved quantities along null geodesics}. We check computationally
that the magnetic part $H_{ab}$ of the Weyl tensor vanishes whereas the shear
$\sigma_{ab}$ and the electric part $E_{ab}$ share a common eigenframe
irrespective of the fluid interpretation of the models. A side result is the
fact that the spacetimes are foliated by a set of \emph{conformally flat }
$3-$dimensional \emph{timelike} slices when the anisotropy of the
\emph{flux-free} fluid is described only in terms of the 3 \emph{principal
inhomogeneous "pressures"} $p_{\alpha}$ or equivalently when the Ricci tensor
shares the same basis of eigenvectors with $\sigma_{ab}$ and $E_{ab}$. The
conformal flatness also indicates that a \emph{10-dimensional algebra} of ICVFs
$\mathbf{\Xi}$ acting on the $3-$dimensional timelike slices is highly possible
to exist enriching in that way the set of conserved quantities admitted by the
SI models found in the present paper.
