  As a difference with the positive-definite Riemannian case, in the Lorentzian
case there exists proper second-order symmetric spacetimes, i.e., those with
vanishing second covariant derivative of the Riemannian tensor
($R_{\lambda\mu\nu\rho;\alpha;\beta}=0$) which are not locally symmetric
($R_{\lambda\mu\nu\rho;\alpha}\neq 0$). In fact, they lie in the class of
Brinkmann metrics, and include some known plane waves. Our purpose is to
classify these spacetimes in the 4 dimensional case.
