  On the basis of systematic calculations for 1364 heavy and superheavy nuclei,
including odd-systems, we have found a few candidates for high-K ground states
in superheavy nuclei. The macroscopic-microscopic model based on the deformed
Woods-Saxon single particle potential which we use offers a reasonable
description of SH systems, including known: nuclear masses,
$Q_{\alpha}$-values, fission barriers, ground state deformations, super- and
hyper-deformed minima in the heaviest nuclei. %For odd and odd-odd systems,
both ways of including pairing correlations, % blocking and the quasi-particle
method, have been applied. Exceptionally untypical high-K intruder contents of
the g.s. found for some nuclei accompanied by a sizable excitation of the
parent configuration in daughter suggest a dramatic hindrance of the
$\alpha$-decay. Multidimensional hyper-cube configuration - constrained
calculations of the Potential Energy Surfaces (PES's) for one especially
promising candidate, $^{272}$ Mt, shows a $\backsimeq$ 6 MeV increase in the
fission barrier above the configuration- unconstrained barrier. There is a
possibility, that one such high-K ground- or low-lying state may be the longest
lived superheavy isotope.
