Pseudo-Riemannian and Hessian geometry related to Monge-Ampère structures

We study properties of pseudo-Riemannian metrics corresponding to Monge-Amp\`ere structures on four-dimensional $T^*M$. We describe a family of Ricci flat solutions, which are parametrized by six coefficients satisfying the Pl\"ucker embedding equation. We also focus on pullbacks of the pseudo-metrics on two-dimensional $M$ and describe the corresponding Hessian structures.