  In this paper a Galoisian approach to build propagators through Riccati
equations is presented. The main result corresponds to the relationship between
the Galois integrability of the linear Schr\"odinger equation and the virtual
solvability of the differential Galois group of its associated characteristic
equation. As main application of this approach we solve the Ince's differential
equation through Hamiltonian Algebrization procedure and Kovacic Algorithm to
find the propagator for a generalized harmonic oscillator that has applications
describing the process of degenerate parametric amplification in quantum optics
and the description of the light propagation in a nonlinear anisotropic
waveguide. Toy models of propagators inspired by integrable Riccati equations
and integrable characteristic equations are also presented.
