  In this document we prove higher equations of motion at the level $2$ for
boundary Liouville Conformal Field Theory. As a corollary we present a new
derivation of the Belavin-Polyakov-Zamolodchikov differential equations. Our
method of proof does not rely on the mating of trees machinery but rather
exploits the symmetries of the model through the Ward identities it satisfies.
  To do so we provide a definition of derivatives of the correlation functions
with respect to a boundary insertion which was lacking in the existing
literature, and introduce a new notion of descendant fields related to these
Ward identities.
