The paper under review deals with the solvability of a class of vector fields on the 3-dimensional torus $\Bbb{T}^3$. A vector field in this class has the form $$ L=\frac{\partial}{\partial t}+(a(x)+ib(x))\left(\frac{\partial}{\partial x}+\lambda\frac{\partial}{\partial x}\right), $$ where $(t,x,y)$ are the angular coordinates in $\Bbb{T}^3$, $\lambda\in\Bbb{R}$, and $a, b \in C^\infty(\Bbb{T}^1,\Bbb{R})$. The operator $L$ is globally solvable on $\Bbb{T}^3$ if $L(C^\infty(\Bbb{T}^3))$ is a closed subspace of $C^\infty(\Bbb{T}^3)$. It should be noted that local solvability of linear partial differential operators of principal type is a classical result, and the Nirenberg-Trèves Condition (P) is necessary and sufficient for local solvability [see L. Nirenberg and F. Trèves, Comm. Pure Appl. Math. {\bf 16} (1963), 331--351; MR0163045]. However Condition (P) is not necessary for global solvability and other types of obstructions to solvability need to be taken into account. As mentioned in the introduction of this paper, the authors explore ``how far from Condition (P) we can find global solvability for vector fields''. \par Building on their previous results with A. Kirilov about solvability on tori [J. Pseudo-Differ. Oper. Appl. {\bf 6} (2015), no.~3, 341--360; MR3385200], the authors use Fourier series techniques, together with diophantine type conditions, and the interplay between orders of vanishing of $a$ and $b$ to give conditions for or against global solvability. They prove, in particular, that if $\lambda$ is not a Liouville number; $a+ib$ vanishes only to finite orders; and at each zero the orders of $a$ and $b$ satisfy $2\le \roman{ord}(b) < 2\, \roman{ord}(a)-1$, then $L$ is globally solvable if and only if $b$ changes sign at most once between two consecutive zeros.