  The reflection operators are the simplest examples of the non-local integrals
of motion, which appear in many interesting problems in integrable CFT. For the
so-called Fateev, quantum AKNS, paperclip and KdV integrable structures, they
are built from the (chiral) reflection S-matrices for the Liouville and cigar
CFTs. Here we give the full spectrum of the reflection operators associated
with these integrable structures. We also obtained a relation between the
reflection S-matrices of the cigar and Liouville CFTs. The results of this work
are applicable for the description of the scaling behaviour of the Bethe states
in exactly solvable lattice systems and may be of interest to the study of the
Generalized Gibbs Ensemble associated with the above mentioned integrable
structures.
