  We characterize under what conditions $n\times n$ Hermitian matrices $A_1$
and $A_2$ have the property that the spectrum of $\cos t A_1 + \sin t A_2$ is
independent of $t$ (thus, the trigonometric pencil $\cos t A_1 + \sin t A_2$ is
isospectral). One of the characterizations requires the first $\lceil
\frac{n}{2} \rceil$ higher rank numerical ranges of the matrix $A_1+iA_2$ to be
circular disks with center 0. Finding the unitary similarity between $\cos t
A_1 + \sin t A_2$ and, say, $A_1$ involves finding a solution to Lax's
equation.
