  This paper pursues the study of the Calabi-Yau equation on certain symplectic
non-Kaehler 4-manifolds, building on a key example of Tosatti-Weinkove in which
more general theory had proved less effective. Symplectic 4-manifolds admitting
a 2-torus fibration over a 2-torus base are modelled on one of three solvable
Lie groups. Having assigned an invariant almost-Kaehler structure and a volume
form that effectively varies only on the base, one seeks a symplectic form with
this volume. Our approach simplifies the previous analysis of the problem, and
establishes the existence of solutions in various other cases.
