We derive cubic interaction vertices for a class of higher-derivative
theories involving three arbitrary integer spin fields. This derivation uses
the requirement of closure of the Poincar\`e algebra in four-dimensional flat
spacetime. We find two varieties of permitted structures at the cubic level and
eliminate one variety, which is proportional to the equations of motion, using
suitable field redefinitions. We then consider soft theorems for field theories
with higher-derivative interactions and construct amplitudes in these theories
using the inverse-soft approach.Comment: 18 page