  We classify the (finite and infinite) virtually cyclic subgroups of the pure
braid groups $P_{n}(RP^2)$ of the projective plane. The maximal finite
subgroups of $P_{n}(RP^2)$ are isomorphic to the quaternion group of order 8 if
$n=3$, and to $\Z_{4}$ if $n\geq 4$. Further, for all $n\geq 3$, up to
isomorphism, the following groups are the infinite virtually cyclic subgroups
of $P_{n}(RP^2)$: $\Z$, $\Z_{2} \times \Z$ and the amalgamated product $\Z_{4}
\ast_{\Z_{2}} \Z_{4}$.
