  Starting with a first order in derivatives self-dual model which describes a
massive spin-4 mode in $D=2+1$ dimensions, we have obtained a sequence of three
more new descriptions, which then give us an interconnected self-dual chain
$SD(i)$ with $i=1,2,3,4$ indicating the order in derivatives. We have
demonstrated that a powerful notation in terms of a self-adjoint operator
$\Omega$ in the frame-like scenario truly simplifies the investigation for new
models and at the third order level can be converted to a geometrical
description in terms of the much more usual totally symmetric double traceless
field.
