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

Spectrum of the reflection operators in different integrable structures