  We give a direct proof of the relation between vacuum singular vectors and
conservation laws for the quantum KdV equation or equivalently for
$\Phi_{(1,3)}$-perturbed conformal field theories. For each degree at which a
classical conservation law exists, we find a quantum conserved quantity for a
specific value of the central charge. Various generalizations ($N=1,2$
supersymmetric, Boussinesq) of this result are presented.
