  A previous paper studied the so-called borderline curves of the
Kac--Murdock--Szeg\H{o} matrix
$K_{n}(\rho)=\left[\rho^{|j-k|}\right]_{j,k=1}^{n}$, where $\rho\in\mathbb{C}$.
These are the level curves (contour lines) in the complex-$\rho$ plane on which
$K_n(\rho)$ has a type-1 or type-2 eigenvalue of magnitude $n$, where $n$ is
the matrix dimension. Those curves have cusps at all critical points
$\rho=\rho_c$ at which multiple (double) eigenvalues occur. The present paper
determines corresponding curves pertaining to eigenvalues of magnitude $N\ne
n$. We find that these curves no longer present cusps; and that, when $N<n$,
the cusps have in a sense transformed into loops. We discuss the meaning of the
winding numbers of our curves. Finally, we point out possible extensions to
more general matrices.
