  We investigate the eigenvalue curves of 1-parameter hermitean and general
complex or real matrix flows $A(t)$ in light of their geometry and the uniform
decomposability of $A(t)$ for all parameters $t$. The often misquoted and
misapplied results by Hund and von Neumann and by Wigner for eigencurve
crossings from the late 1920s are clarified for hermitean matrix flows $A(t) =
(A(t))^*$. A conjecture on extending these results to general non-normal or
non-hermitean 1-parameter matrix flows is formulated and investigated. An
algorithm to compute the block dimensions of uniformly decomposable hermitean
matrix flows is described and tested. The algorithm uses the ZNN method to
compute the time-varying matrix eigenvalue curves of $A(t)$ for $t_o \leq t\leq
t_f$. Similar efforts for general complex matrix flows are described. This
extension leads to many new and open problems. Specifically, we point to the
difficult relationship between the geometry of eigencurves for general complex
matrix flows $A(t)$ and a general flow's decomposability into blockdiagonal
form via one fixed unitary or general matrix similarity for all parameters $t$.
