Higher equations of motion at level 2 in Liouville CFT

We prove conjectures of Zamolodchikov and Belavin-Belavin in Liouville conformal field theory (CFT), which are generalisations of the celebrated Belavin-Polyakov-Zamolodchikov equations known as the higher equations of motion. Algebraically, these equations give examples of non-zero singular states in Virasoro modules, which is a relatively rare phenomenon in the physical study of CFT. In probability theory, these equations and their variants have been instrumental in the rigorous derivation of the structure constants of Liouville CFT in the unit disc. The proof builds on a previous work of ours studying the analytic continuation of the Poisson operator of Liouville theory. The main novelty is that this operator admits poles on the Kac table, and the higher equations of motions are obtained via a residue computation.