  We generalize the conformally invariant topological quantum mechanics of a
particle propagating on a punctured plane by introducing a potential that
breaks both the rotational and the conformal invariance down to a ${\bf Z}_2$
angular-dependent discrete symmetry. We derive a topological quantum mechanics
whose localization gauge functions give interesting self-dual equations. The
model contains an order parameter and exhibits a spontaneous symmetry breaking
with two ground states above a critical scale. Unlike the ordinary
$O(2)$-invariant Higgs potential, an angular-dependence is found and saddle
points, instead of local maxima, appear, posing subtle questions about the
existence of instantons. The supersymmetric quantum mechanical model is
constructed in both the path integral and the operatorial frameworks.
