  Local quartic interaction of higher-spin gauge field with a scalar field is
considered. In this special case, the nontrivial task of construction of
interacting Lagrangian for the higher spin field in physical gauge was solved
using the full power of Noether's procedure. As a result, the linear on-field
gauge transformation is obtained and the corresponding commutator of
transformation is analyzed. To understand the closure of this algebra the
right-hand side of this commutator is classified in respect to gauge
transformations coming from cubic interactions with different higher spin
symmetric tensor fields and with mixed symmetry tensor fields transformations.
