Summary: The stability of the Nyström method for the Sherman-Lauricella equation on contours with corner points \(c_j\), \(j=0,1,\dots,m\), relies on the invertibility of certain operators \(A_{c_j}\) belonging to an algebra of Toeplitz operators. The operators \(A_{c_j}\) do not depend on the shape of the contour, but on the opening angle \(\theta_j\) of the corresponding corner \(c_j\) and on parameters of the approximation method mentioned. They have a complicated structure and there is no analytic tool to verify their invertibility. To study this problem, the original Nyström method is applied to the Sherman-Lauricella equation on a special model contour that has only one corner point with varying opening angle \(\theta_j\). In the interval \((0.1\pi,1.9\pi)\), it is found that there are 8 values of \(\theta_j\) where the invertibility of the operator \(A_{c_j}\) may fail, so the corresponding original Nyström method on any contour with corner points of such magnitude cannot be stable and requires modification.