We shall present a pseudodifferential calculus modelled on second order but not necessarily elliptic operators. This calculus is a generalization of the one developed by W. Craig, T. Kappeler and W. Strauss in the elliptic setting. It arises naturally when trying to compute an integrating factor to avoid first order perturbations. This reduction turns out to be fundamental to solve the initial value problem for quasilinear non-elliptic Schrödinger equations.