  Determining the equilibrium configuration of an elastic M\"{o}bius band is a
challenging problem. In recent years numerical results have been obtained by
other investigators, employing first the Kirchhoff theory of rods and later the
developable, ruled-surface model of Wunderlich. In particular, the strategy
employed previously for the latter does not deliver an unconstrained
equilibrium configuration for the complete strip. Here we present our own
systematic approach to the same problem for each of these models, with the
ultimate goal of assessing the stability of flip-symmetric configurations. The
presence of pointwise constraints considerably complicates the latter step. We
obtain the first stability results for the problem, numerically demonstrating
that such equilibria render the total potential energy a local minimum. Along
the way we introduce a novel regularization for the for the singular Wunderlich
model that delivers unconstrained equilibria for the complete strip, which can
then be tested for stability.
