  The second order Killing and conformal tensors are analyzed in terms of their
spectral decomposition, and some properties of the eigenvalues and the
eigenspaces are shown. When the tensor is of type I with only two different
eigenvalues, the condition to be a Killing or a conformal tensor is
characterized in terms of its underlying almost-product structure. A canonical
expression for the metrics admitting these kinds of symmetries is also
presented. The space-time cases 1+3 and 2+2 are analyzed in more detail.
Starting from this approach to Killing and conformal tensors a geometric
interpretation of some results on quadratic first integrals of the geodesic
equation in vacuum Petrov-Bel type D solutions is offered. A generalization of
these results to a wider family of type D space-times is also obtained. A
generalization of these results to a wider family of type D space-times is also
obtained.
