N-dimensional geometries and Einstein equations from systems of PDE's
  The aim of the present work is twofold: first, we show how all the $n$-dimensional Riemannian and Lorentzian metrics can be constructed from a certain class of systems of second-order PDE's which are in duality to the Hamilton-Jacobi equation and second we impose the Einstein equations to these PDE's. 
Emanuel Gallo, Magdalena Marciano-Melchor, Gilberto Silva-Ortigoza