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$\mathrm{𝑆𝑒𝑐𝑜𝑛𝑑}-\mathrm{𝑜𝑟𝑑𝑒𝑟𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑖𝑐𝐿𝑜𝑟𝑒𝑛𝑡𝑧𝑖𝑎𝑛𝑠𝑝𝑎𝑐𝑒𝑠}$, that is to say, Lorentzian manifolds with vanishing second derivative $\nabla \nabla R\equiv 0$ of the curvature tensor $R$, are characterized by several geometric properties, and explicitly presented. Locally, they are a product $M=M_1\times M_2$ 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., $\nabla R\ne 0$ at some point), the curvature tensor...