  New explicit solutions to the incompressible Navier-Stokes equations in
$\mathbb{R}^{2}\setminus\left\{ \boldsymbol{0}\right\}$ are determined, which
generalize the scale-invariant solutions found by Hamel. These new solutions
are invariant under a particular combination of the scaling and rotational
symmetries. They are the only solutions invariant under this new symmetry in
the same way as the Hamel solutions are the only scale-invariant solutions.
While the Hamel solutions are parameterized by a discrete parameter $n$, the
flux $\Phi$ and an angle $\theta_{0}$, the new solutions generalize the Hamel
solutions by introducing an additional parameter $a$ which produces a rotation.
The new solutions decay like $\left|\boldsymbol{x}\right|^{-1}$ as the Hamel
solutions, and exhibit spiral behavior. The new variety of asymptotes induced
by the existence of these solutions further emphasizes the difficulties faced
when trying to establish the asymptotic behavior of the Navier-Stokes equations
in a two-dimensional exterior domain or in the whole plane.
