  In this paper we study the Yang-Baxter integrable structure of Conformal
Field Theories with extended conformal symmetry generated by the W_3 algebra.
We explicitly construct various T- and Q-operators which act in the irreducible
highest weight modules of the W_3 algebra. These operators can be viewed as
continuous field theory analogues of the commuting transfer matrices and
Q-matrices of the integrable lattice systems associated with the quantum
algebra U_q(\hat{sl}(3)). We formulate several conjectures detailing certain
analytic characteristics of the Q-operators and propose exact asymptotic
expansions of the T- and Q-operators at large values of the spectral parameter.
We show, in particular, that the asymptotic expansion of the T-operators
generates an infinite set of local integrals of motion of the W_3 CFT which in
the classical limit reproduces an infinite set of conserved Hamiltonians
associated with the classical Boussinesq equation. We further study the vacuum
eigenvalues of the Q-operators (corresponding to the highest weight vector of
the W_3 module) and show that they are simply related to the expectation values
of the boundary exponential fields in the non-equilibrium boundary affine Toda
field theory with zero bulk mass.
