This paper deals with the asymptotics of the integrable two-dimensional flow X=X_{fg} on the torus T_2 solution to the ODEs system: x'=f(x,y), y'=g(x,y), which has a regular first integral h in T_2. Under the min-max condition satisfied by h: min_x max_y h(x,y) < max_x min_y h(x,y) or min_y max_x h(x,y) < max_y min_x h(x,y) we prove that the coordinate x or the coordinate y of the flow X_{fg} is bounded. Restricting ourselves to the three-dimensional flow flow X=X_{ab} x'=f(x,y):=b'(y), y'=g(x,y):=b'(y), z'=h(x,y):=b(y)-a(x), it turns out that both coordinates x and y are bounded when ||a||_∞ = ||b||_∞, and that the three-dimensional flow Herman rotation set of the flow is reduced to a closed line segment. When ||a||_∞ ≠ ||b||_∞, assuming some condition on the roots of a',b', we prove that the Herman rotation set of the flow is planar. Applying the previous results to the Arnold-Beltrami-Children flow with parameters with A=0 and B,C≠0, we show that its Herman rotation set contains a ``fusiform'' shape.