Equality Conditions for the Singular Values of $3\times 3$ Matrices with One-Point Spectrum

Let $\Gamma_a$ be an upper triangular $3\times 3$ matrix with diagonal entries equal to a complex scalar $a$. Necessary and sufficient conditions are found for two of the singular values of $\Gamma_a$ to be equal. These conditions are much simpler than the equality $\operatorname{discr}\varphi=\nobreak 0$, where the expression in the left-hand side is the discriminant of the characteristic polynomial $\varphi$ of the matrix $G_a=\Gamma_a^*\Gamma_a$. Understanding the behavior of singular values of $\Gamma_a$ is important in the problem of finding a matrix with a triple zero eigenvalue that is closest to a given normal matrix $A$.