  In this work, using differential Galois theory, we study the spectral problem
of the one-dimensional Schr\"odinger equation for rational time dependent KdV
potentials. In particular, we compute the fundamental matrices of the linear
systems associated to the Schr\"odinger equation. Furthermore we prove the
invariance of the Galois groups with respect to time, to generic values of the
spectral parameter and to Darboux transformations.
