  In this paper we continue the study of bi-conformal vector fields started in
{\em Class. Quantum Grav.} {\bf 21} 2153-2177. These are vector fields defined
on a pseudo-Riemannian manifold by the differential conditions $\lie
P_{ab}=\phi P_{ab}$, $\lie\Pi_{ab}=\chi\Pi_{ab}$ where $P_{ab}$, $\Pi_{ab}$ are
orthogonal and complementary projectors with respect to the metric tensor
$\rmg_{ab}$ and $\lie$ is the Lie derivative. In a previous paper we explained
how the analysis of these differential conditions enabled us to derive local
geometric characterizations of the most relevant cases of {\em conformally
separable} (also called double twisted) pseudo Riemannian manifolds. In this
paper we carry on this analysis further and provide local invariant
characterizations of conformally separable pseudo-Riemannian manifolds with
{\em conformally flat} leaf metrics. These characterizations are rather similar
to that existing for conformally flat pseudo-Riemannian manifolds but instead
of the Weyl tensor, we must demand the vanishing of certain four rank tensors
constructed from the curvature of an affine, non-metric, connection
(bi-conformal connection). We also speculate with possible applications to
finding results for the existence of foliations by conformally flat
hypersurfaces in any pseudo-Riemannian manifold.
