On a theorem of Kara and Klyachko

There exists a finite set of pairs $\left({\Gamma }_0\right(N){\tau }_1,{\Gamma }_0(N\left){\tau }_2\right)$ of points of the modular curve $X_0\left(N\right)$ with ${\Gamma }_0\left(N\right){\tau }_1\ne {\Gamma }_0\left(N\right){\tau }_2$ but $\left(j\right({\tau }_1),j(N{\tau }_1\left)\right)=\left(j\right({\tau }_2),j(N{\tau }_2\left)\right)$, which are the singularities of the plane model $Z_0\left(N\right)$ of $X_0\left(N\right)$. A theorem of Kara and Klyachko gives a characterization of these pairs of points of $X_0\left(N\right)$. However, their proof for this theorem contains an erroneous assertion. Following their idea, we give an elementary proof for this theorem in details.