  We study homogeneous Einstein metrics on indecomposable non-K\"ahlerian
C-spaces, i.e. even-dimensional torus bundles $M=G/H$ with $\mathsf{rank}
G>\mathsf{rank} H$ over flag manifolds $F=G/K$ of a compact simple Lie group
$G$. Based on the theory of painted Dynkin diagrams we present the
classification of such spaces. Next we focus on the family \[ M_{\ell, m,
n}:=\mathsf{SU}(\ell+m+n)/\mathsf{SU}(\ell)\times\mathsf{SU}(m)\times\mathsf{SU}(n)\,,\quad
\ell, m, n\in\mathbb{Z}_{+} \] and examine several of its geometric properties.
We show that invariant metrics on $M_{\ell, m, n}$ are not diagonal and beyond
certain exceptions their parametrization depends on six real parameters. By
using such an invariant Riemannian metric, we compute the diagonal and the
non-diagonal part of the Ricci tensor and present explicitly the algebraic
system of the homogeneous Einstein equation. For general positive integers
$\ell, m, n$, by applying mapping degree theory we provide the existence of at
least one $\mathsf{SU}(\ell+m+n)$-invariant Einstein metric on $M_{\ell, m,
n}$. For $\ell=m$ we show the existence of two $\mathsf{SU}(2m+n)$ invariant
Einstein metrics on $M_{m, m, n}$, and for $\ell=m=n$ we obtain four
$\mathsf{SU}(3n)$-invariant Einstein metrics on $M_{n, n, n}$. We also examine
the isometry problem for these metrics, while for a plethora of cases induced
by fixed $\ell, m, n$, we provide the numerical form of all non-isometric
invariant Einstein metrics.
