Extrinsically homogeneous Lagrangian submanifolds of the pseudo-nearly K\"ahler SL(2,R)(2,R)

We consider the pseudo-nearly K\"ahler SL(2,R)(2,R) and we study its Lagrangian submanifolds. We provide examples of Lagrangian submanifolds which do not have an analogue in S³³. We also provide an expression for the isometry group of SL(2,R)(2,R) with the pseudo-Riemannian nearly K\"ahler metric. The main result is a complete classification of extrinsically homogeneous Lagrangian submanifolds in this space.