This paper deals with the asymptotic analysis of the family of the three-dimensional flows X on the torus, solutions to the ODE's system x'=f(x,y), y'=g(x,y), z'=h(x,y) in [0,∞), where f,g,h are regular periodic fonctions satisfying the condition f ∂x h + g ∂yh = 0. Under the min-max condition satisfied by the first integral h for the flow min_x max_y h(x,y) ≠ max_x min_y h(x,y), we prove that one of the coordinates x or y of the flow is bounded in [0,∞). Restricting ourselves to the subclass of flows defined by f(x,y):=b'(y), g(x,y):=b'(y), h(x,y):=b(y)-a(x), it turns out that both coordinates x and y are bounded when ||a||_∞ = ||b||_∞, and that the Herman rotation set of the flow is then reduced to a closed line segment. When ||a||∞ ≠ ||b||∞, assuming some extra condition on the roots of the derivatives a',b', we prove that the Herman rotation set is planar and contains non-degenerate triangles. Finally, we refine this result in the case of the Arnold-Beltrami-Childress flow ABC with A=0, by showing that the planar Herman rotation set contains a fusiform shape.