= Solution
For the <rotating-frame normalization of resonant amplitude forcing>, use a rotating phase, $A=B\exp[i(\widetilde\Omega T+\delta)]$. The real $\delta$ must occur inside the factor of $i$; the printed change of variables omits that $i$ on the phase constant. The correctly phased substitution gives
$$
B_T+i\widetilde\Omega B=\widetilde\mu B-c|B|^2B
+e\,\overline B^{\,2},\qquad e=\widetilde\epsilon>0.
$$
A negative real forcing coefficient can be made positive by changing the forcing phase. Assuming $c>0$, set
$$
B=\frac ec C,\qquad \mathcal T=\frac{e^2}{c}T,\qquad
\mu=\frac{c\widetilde\mu}{e^2},\qquad
\omega=\frac{c\widetilde\Omega}{e^2}.
$$
All three terms then have the same coefficient scale, giving
$$
\boxed{C_{\mathcal T}+i\omega C
=\mu C-|C|^2C+\overline C^{\,2}.}
$$
Renaming $\mathcal T$ as $T$ gives the requested canonical equation. This rescaling preserves forward time. If $c<0$, a forward-time normalization instead leaves a positive cubic term, and if $c=0$ a cubic normalization is impossible. If $e=0$, the unforced <Landau amplitude equation> must be treated separately. Thus the printed canonical form implicitly concerns saturation in a <supercritical bifurcation> with nonzero forcing.
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