We study the two-dimensional Euler flow solution to ∂ t X(•, x) = b(X(•, x)) for x in the torus T 2 := R 2 /2πZ 2 , where b is the vector field defined on T 2 by b(x) = b(x 1 , x 2) := (− A cos x 1 − B sin x 2 , A sin x 1 + B cos x 2) with A, B ∈ R \ {0}. We derive for any x ∈ T 2 , the asymptotics of X(t, x) as t tends to ∞, depending on whether |A| = |B| or |A| = |B|. In the first case, the orbits of the flow are all bounded. In the second case, it turns out that one of the coordinates of X(t, x) is bounded with an explicit bound, while the other one is equivalent to a(x) t. The function a does not vanish in T 2 and satisfies uniform bounds which depend on parameters A, B. When |A| = |B|, we also prove that for any global first integral u of the flow X with a periodic gradient, ∇u has at least a cluster point of roots in T 2. This shows the complexity as well as the interest of this two-dimensional Euler flow.