  In this paper we construct a family of complex analytic manifolds that
generalize Inoue surfaces and Oeljeklaus-Toma manifolds. To a matrix $M$ in
$SL(N,\mathbb{Z})$ satisfying some mild conditions on its characteristic
polynomial we associate a manifold $T(M,\mathbf{D})$ (depending on an auxiliary
parameter $\mathbf{D}$). This manifold fibers over the $s$-dimensional torus
$\mathbb{T}^s$, where $s$ is the number of real eigenvalues of $M$. The fiber
is the $N$-dimensional torus $\mathbb{T}^{N}$, and the monodromy matrices are
certain polynomials of the matrix $M$. The basic difference of our construction
from the preceding ones is that we admit non-diagonalizable matrices $M$ and
the monodromy of the above fibration can also be non-diagonalizable. We prove
that for a large class of non-diagonalizable matrices $M$ the manifold
$T(M,\mathbf{D})$ does not admit any K\"ahler structure and is not homeomorphic
to any of Oeljeklaus-Toma manifolds.
