  The non-linear realisation based on $A_1^{+++}$ is known to describe gravity
in terms of both the graviton and the dual graviton. We extend this analysis at
the linearised level to find the equations of motion for the first higher dual
description of gravity that it contains. We also give a systematic method for
finding the additional fields beyond those in the non-linear realisation that
are required to construct actions for all of the possible dual descriptions of
gravity in the non-linear realisation. We show that these additional fields are
closely correlated with the second fundamental representation of $A_1^{+++}\,$.
