  For complex $\rho$, the spectral properties of the Toeplitz matrix
$K_{n}(\rho)=\left[\rho^{|j-k|}\right]_{j,k=1}^{n}$, often called the
Kac-Murdock-Szeg{\omicron} matrix, have been examined in detail in two recent
papers. The second paper, in particular, introduced the concept of borderline
curves. These are two closed curves in the complex-$\rho$ plane that consist of
all the $\rho$ for which $K_n(\rho)$ possesses some eigenvalue whose magnitude
equals the matrix dimension $n$. The purpose of the present paper is to examine
eigenvalue bifurcations in both a qualitative and a quantitative manner, and to
discuss connections between bifurcations and the borderline curves.
