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 $3 \leq n<29$, a planar lattice $L$ contains an equilateral $n$-gon if and only if $L$ is similar to an integral lattice $L'$ such that $\nu(L')\equiv 3 \pmod 4$ and the largest prime factor $p$ of $\nu(L')$ satisfies $p \leq n$. Moreover, such $L$ contains a convex equilateral $n$-gon, which answers a problem posed by Maehara.