  Second-order symmetric Lorentzian spaces, that is to say, Lorentzian
manifolds with vanishing second derivative of the curvature tensor R, are
characterized by several geometric properties, and explicitly presented.
Locally, they are a product M=M_1 x M_2 where each factor is uniquely
determined as follows: M_2 is a Riemannian symmetric space and M_1 is either a
constant-curvature Lorentzian space or a definite type of plane wave
generalizing the Cahen-Wallach family. In the proper case (i.e., with non-zero
first covariant derivative of R at some point), the curvature tensor turns out
to be described by some local affine function which characterizes a globally
defined parallel lightlike line. As a consequence, the corresponding global
classification is obtained, namely: any complete second-order symmetric space
admits as universal covering such a product M_1 x M_2. From the technical point
of view, a direct analysis of the second-symmetry partial differential
equations is carried out leading to several results of independent interest
relative to spaces with a parallel lightlike vector field ---the so-called
Brinkmann spaces.
