  In this article we continue the study of the geometry of $k$-D'Atri spaces,
$% 1\leq k$ $\leq n-1$ ($n$ denotes the dimension of the manifold)$,$ began by
the second author. It is known that $k$-D'Atri spaces, $k\geq 1,$ are related
to properties of Jacobi operators $R_{v}$ along geodesics, since she has shown
that ${\operatorname{tr}}R_{v}$, ${\operatorname{tr}}R_{v}^{2}$ are invariant
under the geodesic flow for any unit tangent vector $v$. Here, assuming that
the Riemannian manifold is a D'Atri space, we prove in our main result that
${\operatorname{tr}}R_{v}^{3}$ is also invariant under the geodesic flow if $
k\geq 3$. In addition, other properties of Jacobi operators related to the
Ledger conditions are obtained and they are used to give applications to
Iwasawa type spaces. In the class of D'Atri spaces of Iwasawa type, we show two
different characterizations of the symmetric spaces of noncompact type: they
are exactly the $\frak{C}$-spaces and on the other hand they are $k$ -D'Atri
spaces for some $k\geq 3.$ In the last case, they are $k$-D'Atri for all
$k=1,...,n-1$ as well. In particular, Damek-Ricci spaces that are $k$-D'Atri
for some $k\geq 3$ are symmetric.
  Finally, we characterize $k$-D'Atri spaces for all $k=1,...,n-1$ as the $%
\frak{SC}$-spaces (geodesic symmetries preserve the principal curvatures of
small geodesic spheres). Moreover, applying this result in the case of 4%
-dimensional homogeneous spaces we prove that the properties of being a D'Atri
(1-D'Atri) space, or a 3-D'Atri space, are equivalent to the property of being
a $k$-D'Atri space for all $k=1,2,3$.
