  This article gives explicit integral formulas for the so-called generalized
metaplectic operators, i.e. Fourier integral operators (FIOs) of Schr\"odinger
type, having a symplectic matrix as canonical transformation. These integrals
are over specific linear subspaces of R^d, related to the d x d upper left-hand
side submatrix of the underlying 2d x 2d symplectic matrix. The arguments use
the integral representations for the classical metaplectic operators obtained
by Morsche and Oonincx in a previous paper, algebraic properties of symplectic
matrices and time-frequency tools. As an application, we give a specific
integral representation for solutions to the Cauchy problem of Schr\"odinger
equations with bounded perturbations for every instant time t in R, even in the
so-called caustic points.
