  A family of travelling wave solutions to the Fisher-KPP equation with speeds
$c=\pm 5/\sqrt{6}$ can be expressed exactly using Weierstrass elliptic
functions. The well-known solution for $c=5/\sqrt{6}$, which decays to zero in
the far-field, is exceptional in the sense that it can be written simply in
terms of an exponential function. This solution has the property that the
phase-plane trajectory is a heteroclinic orbit beginning at a saddle point and
ends at the origin. For $c=-5/\sqrt{6}$, there is also a trajectory that begins
at the saddle point, but this solution is normally disregarded as being
unphysical as it blows up for finite $z$. We reinterpret this special
trajectory as an exact sharp-fronted travelling solution to a
\textit{Fisher-Stefan} type moving boundary problem, where the population is
receding from, instead of advancing into, an empty space. By simulating the
full moving boundary problem numerically, we demonstrate how time-dependent
solutions evolve to this exact travelling solution for large time. The
relevance of such receding travelling waves to mathematical models for cell
migration and cell proliferation is also discussed.
