  We review some previous results about the Calabi-Yau equation on the
Kodaira-Thurston manifold equipped with an invariant almost-Kaehler structure
and assuming the volume form invariant by the action of a torus. In particular,
we observe that under some restrictions the problem is reduced to a
Monge-Amp\`ere equation by using the ansatz $\tilde \omega=\Omega-dJdu+da$,
where $u$ is a $T^2$-invariant function and $a$ is a $1$-form depending on $u$.
Furthermore, we extend our analysis to non-invariant almost-complex structures
by considering some basic cases and we finally take into account a
generalization to higher dimensions.
