For Lorentzian 2-manifolds $(��_1,g_1)$ and $(��_2,g_2)$ we consider the two product para-K��hler structures $(G^��,J,��^��)$ defined on the product four manifold $��_1\times��_2$, with $��=\pm 1$. We show that the metric $G^��$ is locally conformally flat (resp. Einstein) if and only if the Gauss curvatures $��_1,��_2$ of $g_1,g_2$, respectively, are both constants satisfying $��_1=-����_2$ (resp. $��_1=����_2$). We give the conditions on the Gauss curvatures for which every Lagrangian surface with parallel mean curvature vector is the product $��_1\times��_2\subset��_1\times��_2$, where $��_1$ and $��_2$ are curves of constant curvature. We study Lagrangian surfaces in the product $d{\mathbb S}^2\times d{\mathbb S}^2$ with non null parallel mean curvature vector and finally, we explore the stability and Hamiltonian stability of certain minimal Lagrangian surfaces and $H$-minimal surfaces.

11 pages. arXiv admin note: text overlap with arXiv:1305.1561