Higher equations of motion at level 2 in Liouville CFT

In a previous work, we investigated the analytic continuation of the bulk Poisson operator of Liouville conformal field theory on the holomorphic part of Fock space and used it to construct irreducible representations of the Virasoro algebra at the degenerate values of the conformal weights. Here, we study two cases where the Poisson operator admits some simple poles on the Kac table: the bulk Poisson operator on the full Fock space (both holomorphic and anti-holomorphic part), and the boundary Poisson operator on the Fock space. As a consequence, the derivative of top singular vector does not vanish, and we can identify it with a scalar multiple of a primary field of same conformal weight. These are known as higher-equations of motions in physics and have been studied in connection with minimal gravity.