  We present a complete classification of Hausdorff locally compact polycyclic
monoids up to a topological isomorphism. A {\em polycyclic monoid} is an
inverse monoid with zero, generated by a subset $\Lambda$ such that $xx^{-1}=1$
for any $x\in\Lambda$ and $xy^{-1}=0$ for any distinct $x,y\in\Lambda$. We
prove that any non-discrete Hausdorff locally compact topology with continuous
shifts on a polycyclic monoid $M$ coincides with the topology of one-point
compactification of the discrete space $M\setminus\{0\}$.
