We consider the BRST invariant effective action of the non-abelian BF
topological theory in $1+1$ dimensions with gauge group $Sl(2,\mathbb{R})$. By
considering different gauge fixing conditions, the zero-curvature field
equation give rise to several well known integrable equations. We prove that
each integrable equation together with the associated ghost field evolution
equation, obtained from the BF theory, is a BRST invariant system with an
infinite sequence of BRST invariant conserved quantities. We construct
explicitly the systems and the BRST transformation laws for the KdV sequence
(including the KdV, mKdV and CKdV equations) and Harry Dym integrable equation.
