  Achieving exact unidirectional invisibility in a finite frequency band has
been an outstanding problem for many years. We offer a simple solution to this
problem in two dimensions that is based on our solution to another more basic
open problem of scattering theory, namely finding scattering potentials
$v(x,y)$ in two dimensions whose scattering problem is exactly solvable for
energies not exceeding a critical value $E_c$. This extends the notion of
quasi-exact solvability to scattering theory and yields a simple condition
under which the first Born approximation gives the exact expression for the
scattering amplitude whenever the wavenumber for the incident wave is not
greater than $\alpha:=\sqrt{E_c}$. Because this condition only restricts the
$y$-dependence of $v(x,y)$, we can use it to determine classes of such
potentials that have certain desirable scattering features. This leads to a
partial inverse scattering scheme that we employ to achieve perfect broadband
unidirectional invisibility in two dimensions. We discuss an optical
realization of the latter by identifying a class of two-dimensional isotropic
active media that do not scatter incident TE waves with wavenumber in the range
$(\alpha/\sqrt 2,\alpha]$ and source located at $x=\infty$, while scattering
the same waves if their source is relocated to $x=-\infty$.
