Complete classification of second-order symmetric spacetimes

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 ($R_{\lambda \mu \nu \rho ;\alpha }\mathrm{\ne }0$). 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.