  We consider radially twisted nanotubes in the low-energy approximation where
the dynamics is governed by a one-dimensional Dirac equation. The mechanical
deformation of the nanotubes is reflected by the presence of an effective
vector potential. We discuss twisted carbon and boron-nitride nanotubes, where
deformations give rise to periodic and nonperiodic finite-gap Hamiltonians. The
intimate relation of these systems with the integrable
Ablowitz-Kaup-Newell-Segur hierarchy is exploited in the study of their
spectral properties as well as in the computation of the (local) density of
states. We show that a nonlinear hidden supersymmetry generated by local
supercharges arises naturally in the finite-gap configurations of twisted
nanotubes with time-reversal symmetry. The properties of the van Hove
singularities are encoded in its structure.
