  We analyze the eigenvalue problem of a quantum particle on the line with the
generalized pointlike potential of three parameter family. It is shown that the
energy surface in the parameter space has a set of singularities, around which
different eigenstates are connected in the form of paired spiral stairway. An
examplar wave-function aholonomy is displayed where the ground state is
adiabatically turned into the second excited state after cyclic rotation in the
parameter space.
  KEYWORDS: one-dimensional system, $\delta'$ potential, non-trivial topology
in quantum mechanics, exotic wave-function aholonomy
