  Here we demonstrate that the sixth order (in derivatives) spin-3 self-dual
model can be obtained from the fifth order self-dual model via a Noether Gauge
Embedding (NGE) of longitudinal Weyl transformations
$\eta_{(\mu\nu}\partial_{\alpha)}\Phi$. In the case of doublet models we can
show that the massive spin-3 Singh-Hagen theory is dual to a fourth and to a
sixth order theory, via a double round of the NGE procedure by imposing
traceless longitudinal (reparametrization-like) symmetries
$\partial_{(\mu}\tilde{\xi}_{\nu\alpha)}$ in the first round and transverse
Weyl transformations $\eta_{(\mu\nu}\psi^T_{\alpha)}$ in the second one. Our
procedure automatically furnishes the dual maps between the corresponding
fields.
