Planar lattices and equilateral odd-gons

For a planar integral lattice $L$, let $\nu (L)$ denote the square-free part of the integer $D(L{)}^2$, where $D(L)$ stands for the area of a fundamental parallelogram of $L$. For each odd integer $n$ with $3\le n<29$, a planar lattice $L$ contains an equilateral $n$-gon if and only if $L$ is similar to an integral lattice $L^{\mathrm{\prime }}$ such that $\nu (L^{\mathrm{\prime }})\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}4)$ and the largest prime factor $p$ of $\nu (L^{\mathrm{\prime }})$ satisfies $p\le n$. Moreover, such $L$ contains a convex equilateral $n$-gon, which answers a problem posed by Maehara.