  The monodromy matrix, ${\hat{\cal M}}$, is constructed for two dimensional
tree level string effective action. The pole structure of ${\hat{\cal M}}$ is
derived using its factorizability property. It is found that the monodromy
matrix transforms non-trivially under the non-compact T-duality group, which
leaves the effective action invariant and this can be used to construct the
monodromy matrix for more complicated backgrounds starting from simpler ones.
We construct, explicitly, ${\hat{\cal M}}$ for the exactly solvable
Nappi-Witten model, both when B=0 and $B\neq 0$, where these ideas can be
directly checked. We consider well known charged black hole solutions in the
heterotic string theory which can be generated by T-duality transformations
from a spherically symmetric `seed' Schwarzschild solution. We construct the
monodromy matrix for the Schwarzschild black hole background of the heterotic
string theory.
