  We study the Liouville type problem for the stationary 3D Navier-Stokes
equations on $\Bbb R^3$. Specifically, we prove that if $v$ is a smooth
solution to (NS) satisfying $\omega={\rm curl}\,v \in L^q (\Bbb R^3) $ for some
$\frac32 \leq q< 3$, and $|v(x)|\to 0$ as $|x|\to +\infty$, then either $v=0$
on $\Bbb R^3$, or $\int_{\Bbb R^6} \Phi_+ dxdy=\int_{\Bbb R^6} \Phi_-
dxdy=+\infty$, where $\Phi(x,y) :=\frac{1}{4\pi}\frac{\omega
(x)\cdot(x-y)\times (v(y)\times \omega(y) )}{|x-y|^3} $, and $\Phi_\pm:=\max\{
0, \pm \Phi\}$. The proof uses crucially the structure of nonlinear term of the
equations.
