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In 1993, a proof was published, within ``Classical and Quantum Gravity,''that there are no regular solutions to the {\it linearized} version of thetwisting, type-N, vacuum solutions of the Einstein field equations. While thisproof is certainly correct, we show that the conclusions drawn from that factwere unwarranted, namely that this irregularity caused such solutions not to beable to truly describe pure gravitational waves. In this article, we resolvethe paradox---since such first-order solutions must always have singular linesin space for all sufficiently large values of $r$---by showing that if weperturbatively iterate the solution up to the third order in small quantities,there are acceptable regular solutions. That these solutions become flat beforethey become non-twisting tells us something interesting concerning the generalbehavior of solutions describing gravitational radiation from a bounded source.Comment: 11 pages, a plain TeX file, submitted to ``Classical and Quantum  Gravity'

An iterative approach to twisting and diverging, type-N, vacuum Einstein equations: a (third-order) resolution of Stephani's `paradox'