  A group $G$ is said to have the Howson property (or to be a Howson group) if
the intersection of any two finitely generated subgroups of $G$ is finitely
generated subgroup. It is proved that descending HNN-extension is not a Howson
group under some assumptions satisfied by the base group of HNN-extension. In
particular, a result of the paper joined with a Burns - Brunner result
(received in 1979) implies that any descending HNN-extension of non-cyclic free
group does not have the Howson property.
