  Modern applications require a robust and theoretically strong tool for the
realistic modeling of electronic states in low dimensional nanostructures. The
$k \cdot p$ theory has fruitfully served this role for the long time since its
creation. During last two decades several problems have been detected in
connection with the application of the $k \cdot p$ approach to such
nanostructures. These problems are closely related to the violation of the
ellipticity conditions for the underlying system, the fact that until recently
has been largely overlooked. We demonstrate that in many cases the models
derived by a formal application of the Luttinger-Kohn theory fail to satisfy
the ellipticity requirements. The detailed analysis, presented here on an
example of the $6 \times 6$ Hamiltonians, shows that this failure has a strong
impact on the physically important properties conventionally studied with these
models.
