  A five-dimensional (5D) generalized G\"odel-type manifolds are examined in
the light of the equivalence problem techniques, as formulated by Cartan. The
necessary and sufficient conditions for local homogeneity of these 5D manifolds
are derived. The local equivalence of these homogeneous Riemannian manifolds is
studied. It is found that they are characterized by three essential parameters
$k$, $m^2$ and $\omega$: identical triads $(k, m^2, \omega)$ correspond to
locally equivalent 5D manifolds. An irreducible set of isometrically
nonequivalent 5D locally homogeneous Riemannian generalized G\"odel-type
metrics are exhibited. A classification of these manifolds based on the
essential parameters is presented, and the Killing vector fields as well as the
corresponding Lie algebra of each class are determined. It is shown that the
generalized G\"odel-type 5D manifolds admit maximal group of isometry $G_r$
with $r=7$, $r=9$ or $r=15$ depending on the essential parameters $k$, $m^2$
and $\omega$. The breakdown of causality in all these classes of homogeneous
G\"odel-type manifolds are also examined. It is found that in three out of the
six irreducible classes the causality can be violated. The unique generalized
G\"odel-type solution of the induced matter (IM) field equations is found. The
question as to whether the induced matter version of general relativity is an
effective therapy for these type of causal anomalies of general relativity is
also discussed in connection with a recent article by Romero, Tavakol and
Zalaletdinov.
