  We consider the three-dimensional incompressible Euler equation
\begin{equation*}\left\{\begin{aligned} &\partial_t \Omega+U \cdot \nabla
\Omega+\Omega\cdot \nabla U=0 \\ &\Omega(x,0)=\Omega_0(x) \end{aligned}\right.
\end{equation*} in the whole space $\mathbb{R}^3$. Under the assumption that
the initial velocity is helical and without swirl, we prove the global
well-posedness of weak solutions in $L^1_1 \bigcap L^{\infty}_1(\mathbb{R}^3)$.
The vortex transport formula is also obtained in our article.
