  This paper is a direct continuation of\ \BLZ\ where we begun the study of the
integrable structures in Conformal Field Theory. We show here how to construct
the operators ${\bf Q}_{\pm}(\lambda)$ which act in highest weight Virasoro
module and commute for different values of the parameter $\lambda$. These
operators appear to be the CFT analogs of the $Q$ - matrix of Baxter\ \Baxn, in
particular they satisfy famous Baxter's ${\bf T}-{\bf Q}$ equation. We also
show that under natural assumptions about analytic properties of the operators
${\bf Q}(\lambda)$ as the functions of $\lambda$ the Baxter's relation allows
one to derive the nonlinear integral equations of Destri-de Vega (DDV)\ \dVega\
for the eigenvalues of the ${\bf Q}$-operators. We then use the DDV equation to
obtain the asymptotic expansions of the ${\bf Q}$ - operators at large
$\lambda$; it is remarkable that unlike the expansions of the ${\bf T}$
operators of \ \BLZ, the asymptotic series for ${\bf Q}(\lambda)$ contains the
``dual'' nonlocal Integrals of Motion along with the local ones. We also
discuss an intriguing relation between the vacuum eigenvalues of the ${\bf Q}$
- operators and the stationary transport properties in boundary sine-Gordon
model. On this basis we propose a number of new exact results about finite
voltage charge transport through the point contact in quantum Hall system.
