An {\em MEP}.1 plan for a $2^m$ factorial experiment is able to search and estimate one possible nonzero interaction effect along with estimating the general mean and main effects. Construction of such a plan has been tackled by several authors. Some of these works are developed for all $m$, in which are focused on $m=2^h-1$ or $m=4\lambda-1$ for positive integers $h>2$ and $\lambda\geq2$. This leads to a gap for having {\em MEP}.1 plans with well-structured runs set and probably with a reasonable number of treatments for $m\not=2^h-1$ or $4\lambda-1$. In this paper, we characterize and present a series of highly structured {\em MEP}.1 plans for all odd $m=4\lambda\pm1$, $\lambda\geq2$, with $N=2m$ runs. From these designs one can obtain the {\em MEP}.1 plans for even $m$ with $N=2m+1$ treatments. The exemptions are $m=$ 8 and 9 for which we have 1 or 2 extra runs.
