The Calabi-Yau equation for $T^2$-bundles over ${\mathbb{T}}^2$: the non-Lagrangian case

In the spirit of [10,2], we study the Calabi-Yau equation on $T^2$-bundles over ${\mathbb{T}}^2$ endowed with an invariant non-Lagrangian almost-K"ahler structure showing that for $T^2$-invariant initial data it reduces to a Monge-Amp`ere equation having a unique solution. In this way we prove that for every total space $M^4$ of an orientable $T^2$-bundle over ${\mathbb{T}}^2$ endowed with an invariant almost-K"ahler structure the Calabi-Yau problem has a solution for every normalized $T^2$-invariant volume form.