Some Remarks on Systems of Equiangular Lines

In this note, we study the maximum number $N_{\alpha }(d)$ of a system of equiangular lines in ${\mathbb{R}}^d$ with cosine $\alpha$, where $\frac{1}{\alpha }$ is not an odd positive integer. This note is motivated by a remark in a $2018$ paper by Balla, Dr"{a}xler, Keevash and Sudakov.